Chapters 8 & 9

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Z-interval, t-interval, hypothesis testing

What is the difference in assumptions between the one-mean t-test and the one-mean z-test?

The one mean z-test - ϑ is known
The one mean t-test - ϑ unknown

What are the two ways to create a narrower confidence interval?

(1) decrease ϑ
(2) increase n (sample size)

What are the two ways to create a wider confidence interval?

(1) increase ϑ
(2) decrease n (sample size)

Formula for confidence interval, sigma known

What does ∝ signify?

The area in both tails. Usually .10 or less

A sample of 36 soda cans was taken and the average number of ounces was found to be 12.4 oz. (assume ϑ=2). Find a 95% confidence interval for µ, the average number of oz in the can. Interpret your results.

(11.747, 13.053)
We are 95% confident that the average number of ounces in a soda can is between 11.8 ounces and 13.1 ounces.

What determines "z" in the confidence interval formula?

Changing the confidence level.

What is the relationship between confidence (high or low) and the length of a confidence interval?

Higher confidence means a wider interval. Lower confidence means a narrower interval.

A sample of 25 typists has an average typing speed of 85 wpm. Assume ó=19. Find a 99% confidence interval for the average speed of typists. Interpret your results.

(75.2, 94.8)
We are 99% confident the average typing speed for typists is between 75.2 words per minute and 94.8 words per minute.

What calculator function is used to find a confidence interval when ó is known?

STAT TESTS Z-interval

Formula to calculate E (margin of error)

What is the relationship between the margin of error and the length of the confidence interval?

The margin of error is half the length of the confidence interval.

Relate the precision with which x-bar estimates µ with the size of the margin of error.

The more precision with which x-bar estimates µ, the smaller the margin of error.

An AP poll found that 38% of parents said they were unlikely to give permission for their kids to be vaccinated at school (sample of 1003 adults). The margin of sampling error is +- 3.1 percentage points for all adults. What is the confidence interval? What is the length of the confidence interval?

(34.9, 41.1)
Length of confidence interval: 6.2

Formula to find the sample size required for a 1-∝ confidence interval.

The FAA estimated with 90% confidence that the mean flight time from Albuquerque NM to Dallas TX to be between 99 minutes and 107.8 minutes. Assume n = 9 and ϑ = 8. (1) Find the margin of error. (2) Find how many times larger must a sample size be to halve the margin of error.

(1) 4.4
(2) 4 (36/9)

What are the properties of the t-distribution?

a) centered at 0 (like the standard normal distribution)
b) symmetric/bell-shaped
c) wider than the normal distribution
d) area under curve = 1
e) uses degrees of freedom (n-1) to find t-values

Which would result in a wider confidence interval? 90% confidence level or 95% confidence level?

95% confidence level would result in a wider confidence interval. Increasing the confidence level increases the length of the confidence interval.

Which would result in a wider confidence interval?
n=100 or n=400

n=100 would result in a wider confidence interval. Decreasing the sample size increases the length of the confidence interval.

What is the appropriate z-value for a 95% confidence level?

1.96

What is the appropriate z-value for a 90% confidence level?

1.645

What is the appropriate z-value for a 99% confidence level?

2.576

What is the appropriate z-value for an 80% confidence level?

1.282

What is the appropriate z-value for an 85% confidence level?

1.44
(Hint: invnorm (.075))

What is the appropriate z-value for a 50% confidence level?

.674
(Hint: invnorm (.25))

Suppose that a random sample of 50 bottles of a particular brand of cough medicine is selected and the alcohol content of each bottle is determined. Let µ denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the sample of 50 results in a 95% confidence interval for µ of (7.8, 9.4).
(1) Would a 90% confidence interval have been narrower or wider than the given interval?

(1) The 90% confidence interval would have been narrower because decreasing the confidence level from 95% to 90% will decrease the confidence interval. The value of Z∝/2 for a 90% confidence level (1.28) is smaller than the Z∝/2 value for the 95% confidence level (1.64); therefore the confidence interval would be narrower.

Consider the following statement: If the process of selecting a sample of size 50 and then computing the corresponding 95% confidence interval is repeated 100 times, 95 of the resulting interval will include µ. Is this statement correct? Why or why not?

** This statement is correct. In the long run, after computing the corresponding 95% confidence interval many times, 95 of the resulting confidence intervals will include µ.

Two intervals (114.4, 115.6) and (114.1, 115.9) are confidence intervals for µ=true average resonance frequency (in hertz) for all tennis rackets of a certain type. (1) What is the value of the sample mean resonance frequency? (2) The confidence level for one of these intervals is 90% and for the other it is 99%. Which is which, and how can you tell?

(1) x-bar = 115 hertz
(2) the 90% confidence interval is (114.4, 115.6) and the 99% confidence interval is (114.1, 115.9). The confidence interval is larger for the 99% confidence level than for the 90% confidence level.

Five hundred randomly selected working adults living in Calgary, Canada were asked how long, in minutes, their typical daily commute was. The resulting sample mean commute time was 28.5 minutes. Construct and interpret a 90% confidence interval for the mean commute time of working adult Calgary residents. (Assume the population standard deviation is 24.2 minutes).

(26.72, 30.28)
We are 90% confident that the average daily commute time for working adults living in Calgary, Canada, will be between 26.72 minutes and 30.28 minutes.

According to Bride's Magazine, getting married these days can be expensive when all costs are included. A simple random sample of 20 recent U.S. weddings yielded data on wedding costs in dollars (sum of data is $526,538). (1) use the data to obtain a point estimate for the population mean wedding cost, µ, of all recent U.S. weddings. (2) Is your point estimate in part (1) likely to equal µ exactly? Explain your answer.

(1) $26,326.9
(2) No. It is unlikely that a sample mean (x-bar) will exactly equal the population mean, µ. Some sampling error is to be anticipated.

Consumer Reports provides information on new automobile models - including price, mileage ratings, engine size, body size, and indicators of features. A simple random sample of 35 new models yields data on fuel tank capacity, in gallons (sum of data is 664.9 gallons). (1) Find a point estimate for the mean fuel tank capacity of all new automobile models. (2) Determine a 95.44% confidence interval for the mean fuel tank capacity of all new automobile models. Assume ϑ=3.50 gallons. (3) How would you decide whether fuel tank capacities for new automobile models are approximately normally distributed? (4) Must fuel tank capacities for new automobile models be exactly normally distributed for the confidence interval that you obtained in part (2) to be approximately correct? Explain your answer.

(1) x-bar = 18.997
(2) (17.81, 20.18) We can be 95.44% confident that the mean fuel tank capacity of all 2003 automobile models is somewhere between 17.81 and 20.18 gallons.
(3) obtain a normal probability plot of the data.
(4) No, because the sample size is large.

Find the confidence level and ∝ for a 90% confidence interval.

Confidence level = .90
∝ = .10

Find the confidence level and ∝ for a 99% confidence interval.

Confidence level = .99
∝ = .01

What are the assumptions required for using the z-interval procedure?

(1) simple random sample
(2) normal population or large sample (≥30)
(3) sigma (ϑ) known

How important is the normality assumption for the z-interval procedure?

The z-interval procedure works well when the variable is normally distributed and reasonably well if the variable is not normally distributed and the sample size is small or moderate, provided the variable is not too far from being normally distributed.

What is meant by saying that a statistical procedure is "robust"?

A statistical procedure that works reasonably well even when one of its assumptions is violated (or moderately violated) is called a robust procedure relative to that assumption.

Assume that the population standard deviation is known. Is it reasonable to use the z-interval procedure to obtain a confidence interval for the population mean under each of the following circumstances: (1) the sample data contains no outliers, the variable under consideration is roughly normally distributed, and the sample size is 20, (2) the distribution of the variable under consideration is highly skewed and the sample size is 20, (3) the sample data contains no outliers, the sample size is 250, and the variable under consideration is far from being normally distributed.

(1) Reasonable, because of the roughly normal distribution, sample size need not be greater than 30 and outliers do not exist that might call into question the normality assumption.
(2) Not reasonable, because sample size is too small.
(3) Reasonable because of the large sample size.

Of 95% and 99% confidence levels, which will result in the confidence interval's giving a more precise estimate of µ?

The 95% confidence level because decreasing the confidence level improves the precision.

Use the one-mean z-interval procedure to find a confidence interval for the mean of the population from which the sample was drawn:
x-bar = 20
n = 36
ϑ = 3
confidence level = 95%

(19.0, 21.0)

Use the one-mean z-interval procedure to find a confidence interval for the mean of the population from which the sample was drawn:
x-bar = 30
n = 25
ϑ = 4
confidence level = 90%

(28.7, 31.3)

Use the one-mean z-interval procedure to find a confidence interval for the mean of the population from which the sample was drawn:
x-bar = 50
n = 16
ϑ = 5
confidence level = 99%

(46.8, 53.2)

A random sample of 18 venture-capital investments in the fiber optics business sector yielded the following data, in millions of dollars (sum of the data is $113.97 million). (1) determine a 95% confidence interval for the mean amount, µ, of all venture-capital investments in the fiber optics business sector. Assume that the population standard deviation is $2.04 million. (2) Interpret your answer from part (1).

(1) $5.389 million to $7.274 million
(2) We can be 95% confident that the mean amount of all venture-capital investments in the fiber optics business sector is somewhere between $5.389 million and $7.274 million.

Given: safety limit set for cadmium in dry vegetables at 0.5 ppm. Random sample of 12 Bp mushrooms, data obtained sums to 6.31 ppm.
Find and interpret a 99% confidence interval for the mean cadmium level of Bp mushrooms. Assume a population standard deviation of cadmium levels in Bp mushrooms of 0.37 ppm.

0.251 ppm to 0.801 ppm
We can be 99% confident that the mean cadmium level of all Bp mushrooms is somewhere between 0.251 ppm and 0.801 ppm.

According to an article, the mean duration of imprisonment for 32 patients with chronic PTSD was 33.4 months. Assuming that ϑ = 42 months, determine a 95% confidence interval for the mean duration of imprisonment, µ, of all East German political prisoners with chronic PTSD. Interpret your answer in words.

18.8 to 48.0 months
We can be 95% confident that the mean duration of imprisonment, µ, of all East German political prisoners with chronic PTSD is somewhere between 18.8 and 48.0 months.

For the same value of ∝, are t-values greater than or less than z-values?

For the same value of ∝, t-values will be greater than z-values.

Find the area to the right of t=1.771 with df=13.

.05

Find the area to the right of t=2 with df = 30.

Between 0.05 and 0.025

Compare t-curves to the standard normal curve as the number of degrees of freedom becomes larger.

As the number of degrees of freedom becomes larger, t-curves look increasingly like the standard normal curve.

Formula for confidence interval for one population mean when ϑ is unknown.

Standardized version of x-bar

...

Studentized version of x-bar

...

How does the distributions of the standard and studentized versions of x-bar differ?

The studentized version has more spread (wider).

Assumptions for the One-Mean t-Interval Procedure

(1) simple random sample
(2) normal population or large sample
(3) ϑ unknown

Formula for the confidence interval for µ (t-interval procedure)

...

When is a confidence interval exact? When is a confidence interval approximately correct?

The confidence interval is exact for normal populations and is approximately correct for large samples from non-normal populations.

The publication Amusement Business provides figures on the cost for a family of four to spend the day at one of America's Amusement parks. A random sample of 25 families of four that attended amusement parks yielded the following costs, rounded to the nearest dollar (see 8.94 p 391). Obtain and interpret a 95% confidence interval for the mean cost for a family of four to spend the day at an American amusement park. (Note: x-bar=$193.32, s=$26.73).

(182.29, 204.35)
We are 95% confident that the average cost for a family of four to spend the day at an American amusement park will be between $182.29 and $204.35.

null hypothesis

The hypothesis to be tested

alternative hypothesis

The alternative to the null hypothesis

hypothesis test

The problem in a hypothesis test is to decide whether the null hypothesis should be rejected in favor of the alternative hypothesis.

two-tailed test

If the primary concern is deciding whether a population mean, µ, is different from a specified value µ₀, we express the alternative hypothesis as:
H₁: µ ≠ µ₀

left-tailed test

If the primary concern is deciding whether a population mean, µ, is less than a specified value µ₀, we express the alternative hypothesis as:
H₁: µ < µ₀

right-tailed test

If the primary concern is deciding whether a population mean, µ, is greater than a specified value µ₀, we express the alternative hypothesis as:
H₁: µ > µ₀

one-tailed test

A hypothesis test that is either left-tailed or right-tailed.

A snack food company produces a 454 g bag of pretzels and insists that the mean net weight of the bags is 454 g. As part of its program, the quality assurance department periodically performs a hypothesis test to decide whether the packaging machine is working properly, that is, to decide whether the mean net weight of all bags packaged is 454 g. (1) Determine the null hypothesis for the hypothesis test. (2) Determine the alternative hypothesis for the hypothesis test. (3) Classify the hypothesis test as two tailed, left tailed, or right tailed.

(1) H₀: µ = 454 g
(2) H₁: µ ≠ 454 g
(3) two tailed

What mathematical signs are allowed in the null hypothesis?

=, ≤, ≥

What mathematical signs are allowed in the alternative hypothesis?

≠, <, >

The mean charitable contribution per household in the U.S. in 2000 is $1623. A researcher claims that the level of giving has changed since then. State the null and the alternative hypotheses.

H₀: µ = $1623
H₁: µ ≠ $1623

Federal law requires that a jar of peanut butter labeled 32 oz. must contain at least 32 oz. A consumer advocate feels that a certain manufacturer is shorting customers by underfilling jars so that the mean content is less than 32 oz. State the null and alternative hypotheses.

H₀: µ ≥ 32
H₁: µ < 32
left tailed test

A confidence interval for a population mean has a margin of error of 3.4 Determine the length of the confidence interval.

6.8

A confidence interval for a population mean has a margin of error of 3.4. If the sample mean is 52.8, obtain the confidence interval.

(49.4, 56.2)

True or False. The length of a confidence interval can be determined if you know only the margin of error.

True

True or False. The margin of error can be determined if you know only the length of the confidence interval.

True

True or False: The confidence interval can be obtained if you know only the margin of error.

False

True or False: The confidence interval can be obtained if you know only the margin of error and the sample mean.

True

The method for computing the sample size required to obtain a confidence interval with a specified confidence level and margin of error - the number resulting from the formula should be rounded up to the nearest whole number. (1) Why do you want a whole number? (2) Why do you round up instead of down?

(1) The sample size cannot be a fraction.
(2) The result (n) is the smallest value that will provide the required margin of error. If the number were rounded down, the sample size would not be sufficient to ensure the required margin of error.

Infants treated for pulmonary hypertension, called the PH group, were compared with those not so treated, called the control group. One of the characteristics measured was head circumference. The mean head circumference of the 10 infants in the PH group was 34.2 cm. (1) Assuming that head circumferences for infants treated for PH are normally distributed with standard deviation 2.1 cm, determine a 90% confidence interval for the mean head circumference of all such infants. (2) Obtain the margin of error, E, for the confidence interval you found in part (1). (3) Explain the meaning of E in this context in terms of the accuracy of the estimate. (4) Determine the sample size required to have a margin of error of 0.5 cm with a 95% confidence level.

(1) (33.108, 35.292)
(2) E = 1.1 cm
(3) We can be 90% confident that the error made in estimating µ by x-bar is at most 1.1 cm.
(4) 68

Explain the difference in the formulas for the standardized and the studentized version of x-bar.

The denominator of the standardized version of x-bar (z-score) uses the population standard deviation, ϑ, whereas the denominator of the studentized version of x-bar (t-score) uses the sample standard deviation, s.

A variable has a mean of 100 and a standard deviation of 16. 4 observations of this variable have a mean of 108 and a sample standard deviation of 12. Determine the observed value of the:
(1) standardized version of x-bar
(2) studentized value of x-bar

(1) 1
(2) 1.33

Two t-curves have degrees of freedom, 12 and 20, respectively. Which one more closely resembles the standard normal curve? Explain your answer.

The df=20 because as the number of degrees of freedom increases, t-curves look increasingly like the standard normal curve.

According to Scarborough Research, more than 85% of working adults commute by car. Of all U.S. cities, Washington, D.C. and New York City have the longest commute times. A sample of 30 commuters in the Washington, D.C. area yielded the following commute times, in minutes (data set, x-bar=27.97 minutes, s=10.04 minutes). (1) Find a 90% confidence interval for the mean commute time of all commuters in Washington, D.C. (2) Interpret your answer from part (1).

(1) (24.855, 31.085)
(2) We are 90% confident that the average commute time of all commuters in Washington, D.C. is between 24.855 minutes and 31.085 minutes.

A data set gives the additional sleep in hours obtained by a sample of 10 patients using laevohysocyamine hydrobromide (with xbar=2.33 hr, s=2.002 hr). (1) Obtain and interpret a 95% confidence interval for the additional sleep that would be obtained on average for all people using laevohysocyamine hydrobromide. (2) Was the drug effective in increasing sleep? Explain your answer.

(1) 0.90 hr to 3.76 hr
We can be 95% confident that the additional sleep that would be obtained on average for all people using the drug is somewhere between 0.90 hr and 3.76 hr.
(2) It appears so, because, based on the confidence interval, we can be 95% confident that the mean additional sleep is somewhere between 0.90 hr and 3.76 hr and that, in particular, the mean is positive.

Explain the meaning of the term hypothesis as used in inferential statistics.

A hypothesis is a statement that something is true.

Given: safety limit set for cadmium in dry vegetables at 0.5 ppm. A hypothesis test is to be performed to decide whether the mean cadmium level in Bp mushrooms is greater than the government's recommended limit. (1) Determine the null hypotheses, (2) Determine the alternative hypothesis, (3) classify the hypothesis test as two tailed, left tailed, or right tailed.

(1) null hypothesis H₀: µ = .5 ppm
(2) alternative hypothesis H₁: µ > .5 ppm
(3) right tailed test

The recommended dietary allowance (RDA) of iron for adult females under the age of 51 is 18 milligrams (mg) per day. A hypothesis test is to be performed to decide whether adult females under the age of 51 are, on average, getting less than the RDA of 18 mg of iron. (1) Determine the null hypothesis, (2) Determine the alternative hypothesis, (3) classify the hypothesis test as two tailed, left tailed, or right tailed.

(1) null hypothesis H₀: µ = 18 mg
(2) alternative hypothesis H₁: µ < 18 mg
(3) left tailed test

According to the Bureau of Crime Statistics and Research of Australia, the mean length of imprisonment for motor-vehicle theft offenders in Australia is 16.7 months. You want to perform a hypothesis test to decide whether the mean length of imprisonment for motor-vehicle theft offenders in Sydney differs from the national mean in Australia. (1) Determine the null hypothesis, (2) Determine the alternative hypothesis, (3) classify the hypothesis test as two tailed, left tailed, or right tailed.

(1) H₀: µ = 16.7 months
(2) H₁: µ ≠ 16.7 months
(3) two tailed test

test statistic

the z-score (or t-score) that determines if an average is unusual or not

Calculation formula for the test statistic (z-score or t-score)

...

What does the z or t test statistic tell us?

The z or t test statistic tells us how far x-bar is from µ in standard deviations (i.e. the number of standard deviations from the mean). It is the statistic used as a basis for deciding whether the null hypothesis should be rejected.

rejection region

The set of values for the test statistic that lead us to reject H₀ (tail or tails of the distribution)

non-rejection region

The set of values for the test statistic that lead us not to reject H₀

critical values

the boundaries for the rejection/non-rejection regions

Type I Error

Rejecting the null hypothesis when it is in fact true

Type II Error

Not rejecting the null hypothesis when it is in fact false.

Type I Error probability

the probability of a Type I error, denoted ∝, also called the significance level of the hypothesis test

significance level

the probability of making a Type I error, that is, of rejecting a true null hypothesis (denoted ∝)

Type II Error probability

the probability of a Type II error, denoted β - a Type II error occurs if the test statistic falls in the non-rejection region when in fact the null hypothesis is false.

What is the relationship between Type I and Type II Error probabilities?

For a fixed sample size, the smaller we specify the significance level, ∝, the larger will be the probability, β, of not rejecting a false null hypothesis.

Possible conclusions for a hypothesis test

If the null hypotheses is rejected, we conclude that the alternative hypothesis is true. If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alternative hypothesis.

Steps for Hypothesis Tests for One Population Mean when ϑ is known (6)

(1) State the null and alternative hypotheses
(2) Decide on a value for ∝ (significance level)
(3) Compute the test statistic Z
(4) Find the critical values
(5) Conclusion
(6) Interpretation

Calculation of critical values (for hypothesis test for one population when ϑ is known)

+/- Z (∝/2) - two tailed test
- Z ∝ - left tailed test
Z∝ - right tailed test

Suppose a CEO of a company wants to determine whether the average amount of wasted time during an 8-hour day for employees at the company is less than 120 minutes. A random sample of 10 employees gave these results:
108, 131, 112, 113, 117, 113, 130, 105, 111, 128
Assume ϑ = 9.
Do these data provide evidence that the mean wasted time for this company is less than 120 minutes?

Conclusion: Do not reject H₀
Interpretation: There is not enough evidence to conclude that the average amount of wasted time at the company is less than 120 minutes.

True or False: If it is important not to reject a true null hypothesis, the hypothesis test should be performed at a small significance level.

True. The significance level ∝ of a hypothesis test is the probability of making a Type I error (rejecting a true null hypothesis). If this is important, the lower the probability of making such an error the better; thus you should use a small significance level.

True or False: For a fixed sample size, decreasing the significance level of a hypothesis test results in an increase in the probability of making a Type II error.

True. For a fixed sample size, the smaller you specify the significance level ∝, the larger will be the probability β, of not rejecting a false null hypothesis.

Define test statistic

The statistic used as a basis for deciding whether the null hypothesis should be rejected

Define rejection region

The set of values for the test statistic that leads to rejection of the null hypothesis.

Define non-rejection region

The set of values for the test statistic that leads to non-rejection of the null hypothesis.

Define critical values

The values of the test statistic that separate the rejection and non-rejection regions. A critical value is considered part of the rejection region.

Define significance level

The probability of making a Type I error, that is, of rejecting a true null hypothesis.

Identify the two types of incorrect decisions in a hypothesis test. For each incorrect decision, what symbol is used to represent the probability of making that type of error?

(1) Type I - rejecting a true null hypothesis (symbol ∝)
(2) Type II - not rejecting a false null hypothesis (symbol β)

Would it be appropriate to use a t-interval for a sample size of 15? Explain.

It would not be appropriate because the assumption of a normal population or a large sample is not met. We know nothing of the population and the sample is small.

What is the appropriate t-value for the following confidence level and sample size: confidence level 95%, n=17

2.120

What is the appropriate t-value for the following confidence level and sample size: confidence level 90%, n=12

1.796

What is the appropriate t-value for the following confidence level and sample size: confidence level 99%, n=24

2.807

The following data are airborne times for United Airlines flight 448 from Albuquerque to Denver on 10 randomly selected days: 57, 54, 55, 51, 56, 48, 52, 51, 59, 59 . (1) Compute and interpret a 90% confidence interval for the mean airborne time for flight 448. (2) Based on your interval in part (1), if flight 448 is scheduled to depart at 10 a.m., what would you recommend for the published arrival time? Explain.

(1) (52.07, 56.33)
(2) Recommend an arrival time of 10:57 a.m., so that 0% of the flights would be late.

A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate with 95% confidence to within 0.1 lb, the average force required to break the binding? Assume ϑ is known to be 0.8 lb.

246

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose that the specifications state that the mean strength of welds should exceed 100 lb/ in². The inspection team decides to test H₀: µ = 100 versus H₁: µ > 100. Explain why this alternative hypothesis was chosen rather than µ < 100.

The alternative hypothesis was chosen because the mean strength of welds should be greater than 100, thus the alternative hypothesis H₁: µ > 100. The primary concern of the research is to decide whether the population mean is greater than the specified value (and meets specifications).

Does this pair comply with the rules for setting up hypotheses? If not, explain why.
H₀: µ = 15; H₁: µ = 15

Does not comply. H₁ must be stated as ≠ 15, <15, or > 15.

Does this pair comply with the rules for setting up hypotheses? If not explain why.
H₀: µ = 10; H₁: µ > 12

Does not comply. H₁ must use the same number as H₀, the null hypothesis.

Does this pair comply with the rules for setting up hypotheses? If not explain why.
H₀: µ = 123; H₁: µ < 123

Complies

Does this pair comply with the rules for setting up hypotheses? If not explain why.
H₀: µ = 123; H₁: µ = 125

Does not comply. H₁ must use the same number as H₀ and cannot contain the equal sign.

Does this pair comply with the rules for setting up hypotheses? If not explain why.
H₀: µ = 50; H₁: µ ≠ 50

Complies

Researchers have postulated that because of differences in diet, Japanese children have a lower mean blood cholesterol level than U.S. children do. Suppose that the mean level for U.S. children is known to be 170. Let µ represent the true mean blood cholesterol level for Japanese children. What hypothesis should the researchers test? Give the null and alternative hypotheses.

H₀: µ = 170 (the null hypothesis is the hypothesis to be tested)
H₁: µ < 170

Using the test statistic formula for the z-score, determine the required critical value(s) for a right-tailed test with ∝ = 0.05.

1.645

Using the test statistic formula for the z-score, determine the required critical value(s) for a left-tailed test with ∝ = 0.05.

-1.645

Using the test statistic formula for the z-score, determine the required critical value(s) for a two-tailed test with ∝ = 0.05.

-1.96 and +1.96

Decide in the following situations whether the z-test is an appropriate method for conducting
the hypothesis test for a population mean: (a) no outliers, distribution highly skewed, sample size 24, (b) no outliers, mildly skewed, sample size 70

(a) not appropriate
(b) appropriate

Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The Czech and Slovak governments have set a safety limit for cadmium in dry vegetables at 0.5 parts per million (ppm). A random sample of the edible mushroom Boletus pinicola with the resulting data: 0.24, 0.59, 0.62, 0.16, 0.77, 1.33, 0.92, 0.19, 0.33, 0.25, 0.59, 0.32 . At the 5% significance level, do the data provide sufficient evidence to conclude that the mean cadmium level in Boletus pinicola mushrooms is greater than the government's recommended limit of 0.5 ppm? Assume that the population standard deviation of cadmium levels in Boletus pinicola mushrooms is 0.37 ppm. (Note: The sum of the data is 6.31 ppm).

Given: significance level 0.05
ϑ = 0.37
H₀: µ = 0.5 ppm
H₁: µ > 0.5 ppm

Test statistic: z=0.24

Critical value: 1.645

Conclusion: 0.24 is in the non-rejection region

Interpretation: There is not enough evidence to conclude that the mean level of cadmium in Boletus pinicola mushrooms is greater than 0.5 ppm.

Given:
n=45
x-bar = 14.68
ϑ = 4.2
∝ = .01
H₀: µ = 18
H₁: µ < 18

Reject or do not reject null hypothesis?

Test statistic: z = -5.3
Critical value: -2.326
Conclusion: -5.3 is in the rejection region
Interpretation: There is sufficient evidence to conclude that ... the alternative hypothesis is true.

Describe the meaning of P-value of a hypothesis test

To obtain the P-value of a hypothesis test, we assume that the null hypothesis is true and compute the probability of observing a value of the test statistic as extreme as or more extreme than that observed. By extreme we mean "far from what we would expect to observe if the null hypothesis is true." We use the letter P to denote the P-value. The p-value is the area beyond the test statistic in either direction.

If you have a p-value of 0.0168 and a z-score of +/- 2.39, interpret the meaning of these values in context.

The probability of getting a z-score more extreme than +/- 2.39 is 0.0168.

On the calculator (TI-84), how do you find the area to the left of a particular z-score?

NORMALCDF (-1000, Z-score, 0, 1)

On the calculator (TI-84), how do you find the area to the right of a particular z-score?

NORMALCDF (Z-score, 1000, 0, 1)

What p-value(s) describe weak or no evidence against the null hypothesis?

P-value > .10

What p-value(s) describe moderate evidence against the null hypothesis?

.05 < P ≤ .10

What p-value(s) describe strong evidence against the null hypothesis?

.01 < P ≤ .05

What p-value(s) describe very strong evidence against the null hypothesis?

P ≤ .01

How does a large test statistic relate to the area in the tail?

A large test statistic means that there is a smaller area in the tail.

How does a small test statistic relate to the area in the tail?

A small test statistic means that there is a larger area in the tail.

2 methods for determining whether to reject or not reject the null hypothesis

(1) compare the test statistic to the critical values; where the test statistic falls (rejection region or non-rejection region)
(2) compare the p-value to ∝
If the p-value is low, H₀ must go!
If the p-value ≤ ∝, reject H₀
If the p-value > ∝, do not reject H₀

A hot tub manufacturer advertises that with its heating equipment a temperature of 100 degrees F can be achieved in at most 15 minutes. A random sample of 20 tubs is selected and the time needed to reach 100 degrees is determined for each tub. The sample mean is 16 minutes with a standard deviation of 1 minute. Does this information cast doubt on the company's claim? Assume ∝ = 0.01.

(1) H₀: µ ≤ 15
H₁: µ > 15
(2) ∝ = 0.01
(3) Test statistic: t = 4.47
(4) P-value = .000013 (using t-test on calculator)
(5) Compare p-value to ∝
.000013 ≤ 0.01
Reject H₀
(6) There is enough evidence to suggest the average time for a hot tub to reach 100 degrees F is more than 15 minutes (p=.000013, ∝ = 0.01).

An automobile manufacturer who wishes to advertise that one of its models achieves 30 mpg decides to carry out a fuel efficiency test. Six non-professional drivers are selected and each one drives a car from Phoenix to Los Angeles. The resulting fuel efficiencies in mpg are: 27.2, 29.3, 31.2, 28.4, 30.3, 29.6. Assuming that the fuel efficiency is normally distributed, do the data contradict the claim that the true average fuel efficiency is at least 30 mpg? Assume ∝ = 0.05.

(1) H₀: µ ≥ 30 mpg
H₁: µ < 30 mpg
(2) ∝ = 0.05
(3) test statistic t = -1.16
(4) p-value = .1493 (t-test on calculator)
(5) compare p-value to ∝
.1493 ≤ 0.05 ? NO
Do not reject H₀
(6) There is not enough evidence to conclude that the average fuel efficiency in mpg is less than 30 mpg.

For which of the following p-values would the null hypothesis be rejected at a level of ∝ = 0.05:
(a) .001
(b) .021
(c) .078
(d) .047
(e) .148

(a) reject
(b) reject
(c) do not reject
(d) reject
(e) do not reject

State two reasons why including the p-value is prudent when you are reporting the results of a hypothesis test.

(1) it allows you to assess significance at any desired level
(2) it permits you to evaluate the strength of the evidence against the null hypothesis

What is the p-value of a hypothesis test?

The probability of observing a value of the test statistic as extreme or more extreme than that observed. By extreme we mean "far from what we would expect to observe if the null hypothesis is true."

When does the p-value provide evidence against the null hypothesis?

When the p-value is less than or equal to the significance level, ∝

True or False: The p-value is the smallest significance level for which the observed sample data result in rejection of the null hypothesis.

True

The p-value for a hypothesis test is 0.083. For each of the following significance levels, decide whether the null hypothesis should be rejected:
(1) ∝ = 0.05
(2) ∝ = 0.10
(3) ∝ = 0.06

(1) Do not reject (0.083 > 0.05)
(2) Reject (0.083 ≤ 0.10)
(3) Do not reject (0.083 > 0.06)

Determine the strength of the evidence against the null hypothesis:
(a) p = 0.06
(b) p = 0.35
(c) p = 0.027
(d) p = 0.004

(1) moderate
(2) weak or none
(3) strong
(4) very strong

Determine the p-value.
Z = 2.03 right-tailed test

0.0212

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