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Statistics
Statistic 2000 - Test 3 - Section 6.2 + 6.3 + 6.4 + 6.5 + 7.3
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Terms in this set (64)
What requirements are necessary for a normal probability distribution to be a standard normal probability distribution?
The mean and standard deviation have the values of μ=0 and 𝛔= 1.
Which of the following is NOT a descriptor (mô tả) of a normal distribution of a random variable?
The graph is centered (được đặt ở giữa) around 0.
Which of the following groups of terms can be used interchangeably (Hoán đổi cho nhau) when working with normal distributions?
areas, probability, and relative frequencies
A continuous random variable has a _______ distribution if its values are spread evenly over the range of possibilities. (Trải đều trên phạm vi khả năng)
uniform
Which of the following does NOT describe the standard normal distribution?
The graph is uniform.
Finding probabilities associated with distributions that are standard normal distributions is equivalent to _______.
finding the area of the shaded region representing that probability.
The notation P(z<a) denotes _______.
the probability that the z-score is less than a.
Which of the following would be information in a question asking you to find the area of a region under the standard normal curve as a solution?
A distance on the horizontal axis is given
Which of the following is not true?
A z-score is an area under the normal curve.
Where would a value separating the top 15% from the other values on the graph of a normal distribution be found?
the right side of the horizontal scale of the graph
Which of the following statistics are unbiased estimators (ước lượng không thiên vị) of population parameters?
Sample variance used to estimate a population variance.
Sample mean used to estimate a population mean.
Sample proportion used to estimate a population proportion.
_____________ is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.
The sampling distribution (Phân phối mẫu) of a statistic
Which of the following is NOT a property of the sampling distribution of the sample mean?
The distribution of the sample mean tends to be skewed to the right or left.
Which of the following is NOT a property of the sampling distribution of the variance?
The distribution of sample variances tends to be a normal distribution.
_____________ is the distribution of sample proportions, with all samples having the same sample size n taken from the same population.
The sampling distribution of the proportion
Which of the following is a biased estimator? That is, which of the following does not target the population parameter?
Median
The _______ tells us that for a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases.
Central Limit Theorem (Định lý giới hạn trung tâm)
The standard deviation of the distribution of sample means is _______.
Which of the following is NOT a conclusion of the Central Limit Theorem?
The distribution of the sample data will approach a normal distribution as the sample size increases.
The _______ states that if, under a given assumption, the probability of a particular observed event is exceptionally small (such as less than 0.05), we conclude that the assumption is probably not correct.
Rare Event Rule for Inferential (Suy luận) Statistics
The number of _______ for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values.
degrees of freedom
Which of the following is NOT a property of the Student t distribution?
The standard deviation of the Student t distribution is s=1
The _______ is the best point estimate of the population mean.
sample mean
Which of the following is NOT a requirement for constructing a confidence interval for estimating a population mean with σ known?
The confidence level is 95%.
Which of the following would be a correct interpretation of a 99% confidence interval such as 4.1<μ<5.6?
We are 99% confident that the interval from 4.1 to 5.6 actually does contain the true value of μ.
What conditions would produce a negative z-score?
a z-score corresponding to an area located entirely in the left side of the curve
The Standard Normal Distribution
is a normal distribution with the parameters of μ=0 and σ=1. The total area under its density curve is equal to 1
A continuous random variable has a uniform distribution
if its values are spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular (hình hộp chữ nhật) shape.
Properties of Standard Normal Curve
The total area under the curve is 1 (100%).
Mean = 0
Standard Deviation = 1
For the standard normal distribution, a critical value
is a z score separating unlikely values from those that are likely to occur.
The sampling distribution of a statistic (such as a sample mean or sample proportion)
is the distribution (Phân phối) of all values of the statistic when all possible samples of the same size n are taken from the same population.
The sampling distribution of the sample mean
is the distribution of all possible sample means (or the distribution of the variable x¯x¯ x bar ), with all samples having the same sample size n taken from the same population.
The sampling distribution of the sample variance
is the distribution of sample variances, with all samples having the same sample size n taken from the same population.
The sampling distribution of the sample proportion
is the distribution of sample proportions, with all samples having the same sample size n taken from the same population.
In real life, very few distributions have a
mean of 0 and a standard deviation of 1.
To find the area under a normal curve
the only difference is you need to change your mean and standard deviation.
To find the number on the horizontal axis
the only difference is you need to change your mean and standard deviation.
An estimator (ước lượng)
is a statistic used to infer (estimate) the value of a population parameter.
An unbiased estimator
is a statistic that targets the value of the population parameter in the sense that the sampling distribution of the statistic has a mean that is equal to the mean of the corresponding parameter.
biased estimators (Ước tính thiên vị) for the population.
The median, standard deviation, and range
(In other words, if we were to do the same procedure for median, s.d., or range, the answer would not come close to the population parameter.)
When we are using all possible values of our data to find a mean or standard deviation (or median or anything else),
we are finding the population parameter. We can only do this when the population is small enough. Because of this, we still think of it as finding information on a sample. That is why we use s in the calculator instead of 𝛔.
unbiased estimators for the population.
The mean, variance and proportions
(In other words If we take the mean of sample means, it is a good estimator of the population mean. If we take the mean of the sample variances, it is a good estimator of the population variance. If we take the mean of the sample proportions, it is a good estimator of the population proportion.)
Sample Variances
tend to be right skewed.
The Central Limit Theorem
For all samples of the same size n with n>30, the sampling distribution can be approximated by a normal distribution with a mean µ and standard deviation, 𝛔/√𝑛
The original population is not normally distributed but n > 30
As long as there are at least 30 members of the samples involved, the CLT applies, whether the population is normal or not.
The original population is normally distributed
If the original population is known to be normally distributed the Central Limit Theorem automatically applies.
The mean of the sample means
The original population is not normally distributed and n < 30
If the samples are too small and you don't have a normally distributed population, you should not apply the CLT.
The standard deviation of sample mean
When sampling without replacement and the sample size n is greater than 5% of the finite population size N (that is, n>0.05Nn>0.05N bold italic n greater than , 0.05 n )
adjust the standard deviation of sample means σx¯ by multiplying it by the finite population correction factor:
√ (N−n)/(N−1)
the best point estimate for the population mean
The sample mean. In other words, if all you have is one sample, that's the best you can do.
simple random sampling.
You want your sample to be as random as possible. In other words, you want every member of your population to have an equal chance of being part of your sample.
Degrees of Freedom
It is the number of sample values that can vary (khác nhau) after certain restrictions (Hạn chế nhất định) are imposed (Áp đặt). = n - 1.
Standard Normal
*Bell shaped curve
*Useful when you know the mean and standard deviation of a population.
*Z-scores
*Use zinterval in the calculator
Student t
*Bell shaped curve (approximately if the sample size is big enough)
*Useful when you don't know the standard deviation of the population.
*t-scores
*Use tinterval in the calculator
two requirements for using confidence intervals (Khoảng tin cậy)
1. The data must come from a simple random sample.
2. The data should be approximately normal or have a sample size of n>30.
The sample mean x¯
is the best point estimate (or single value estimate) of the population mean μ.
Confidence Interval (Khoảng tin cậy)
x¯ - E < μ < x¯ + E
or: x¯±E
or: (x¯−E,x¯+E)
Student t Distribution
If a population has a normal distribution, then the distribution below is a Student t distribution for all samples of size n. It is commonly referred to simply as a t distribution.
Point estimate of μ
x¯= [(upper confidence limit)+(lower confidence limit)]/2
Margin of error
E = [(upper confidence limit)−(lower confidence limit)]/2
Sample Size
When I am constructing confidence intervals and I do not know the population standard deviation, I use what function of my calculator
Tinterval
The Central Limit Theorem says that if I am given the mean and standard deviation of a sample, and I want to find the probability that mean of the population is less than some number, then in normalcdf
I use the same mean as the sample, but the standard deviation must be divided by the square root of the sample size
Related questions
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Acceptable risk of incorrect rejection is the statistical risk that the auditor has concluded that a population is materially misstated when it is not.
QUESTION
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A bin contains three components from supplier A, four from supplier B, and five from supplier C. If four of the components are randomly selected for testing, what is the probability that each supplier would have at least one component tested?
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(a) determine the null and alternative hypotheses, (b) explain what it would mean to make a Type I error, (c) explain what it would mean to make a Type II error, (d) state the conclusion that would be reached if the null hypothesis is not rejected, and (e) state the conclusion that would be reached if the null hypothesis is rejected. Among all credit cards issued, the proportion of cards that result in default was 0.13 in 2010. A credit analyst with Visa believes this proportion is different today.
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Give a reasonable space S for answer to this question. How much money in coins (not bills) are you carrying?
PROBABILITY
The lifetimes of six major components in a copier are independent exponential random variables with means of 8000, 10,000, 10,000, 20,000, 20,000, and 25,000 hours, respectively. (a) What is the probability that the lifetimes of all the components exceed 5000 hours? (b) What is the probability that none of the components has a lifetime that exceeds 5000 hours? (c) What is the probability that the lifetimes of all the components are less than 3000 hours?