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Chapter 10 AP Stat Confidence intervals
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What is statistical inference?
It provides methods for drawing conclusions about a population from sample data. What is new about formal inferences is that we use probability to express the strength of our conclusions. When you use statistical inference, you are acting as if the data are a random sample or come from a randomized experiment.
In statistics, what is meant by a 95% confidence interval?
A 95% confidence interval means that 95% of the time our interval will capture the population parameter (mean, proportion...). We are 95% confident that __________ lies within our interval.
Does a 95% confidence interval mean there is a 95% probability that the mean is in our interval?
No, either the mean is in the interval or not with a probability 1 or 0. A 95% confidence interval means 95% of the time, the mean from our sample will be in the confidence interval. For this particular sample it is or it is not.
Sketch a label a 95% confidence interval for the standard normal curve.
We should sketch a normal curve with the middle shaded and a small region in both tails, each having 0.025 not shaded.
In a sampling distribution of x-bar, why is the interval of numbers between x-bar+-2s called a 95% confidence interval?
By the empirical rule, a z-score of 2 has area approximately 0.025 above it and a z-score of -2 has area approximately 0.025 below it. A z-score of +-1.96 from the table is more accurate.
Define a level C confidence interval.
A level C confidence interval means for a parameter has two parts:
-An interval calculated from the data, usually of the form: estimate +- margin of error
-A confidence level C, which gives the probability that the interval will capture the true parameter value in reduced samples.
Sketch and label a 90% confidence interval for the standard normal curve.
We should sketch a normal curve with the middle shaded and a small region in both tails, each having area 0.05 not shaded.
What does z* represent?
z
is the value with area C between -z
and z* under the standard normal curve.
What is the value z* for a 95% confidence interval?
z*=1.96
What is the value z* for a 90% confidence interval?
z*=1.645
What is the value z* for a 99% confidence interval?
z*=2.576
What is meant by the upper p critical value of the standard normal distribution?
The number z* with probability p lying to its right under the standard normal curve.
Explain how to find a level C confidence interval for an SRS of size n having unknown mean u and known standard deviation o.
See notes number 13.
What is meant by margin of error?
The margin of error follows from the choice of confidence level. See notes number 14 for full answer.
The margin of error only covers specifically what kind of error?
It only covers random sampling errors.
Why is it best to have high confidence and a small margin of error?
High confidence says that our method almost always gives correct answers. A small margin of error says that we have pinned down the parameter quite precisely.
What happens to the margin of error as z* decreases? Does this result in a higher or lower confidence level?
As z* gets smaller, that means we have LESS confidence. We have a smaller confidence level C. (ex. 90% instead of 95%). It means the margin of error gets SMALLER.
What happens to the margin of error as the standard deviation, o, decreases?
As the standard deviation decreases the margin of error gets smaller.
What happens to the margin of error as n increases? By how many times must the sample size n increase in order to cut the margin of error in half?
As n increases, the margin of error gets SMALLER. To cut the margin of error in half, the sample size must be four times as large.
Be able to rearrange the formula used to determine the sample size n that will yield a confidence interval for a population mean with a specified margin of error m in order to solve for n.
See notes number 19.
Are assumptions important to statistics? Why?
Yes, because if we do not determine that our data satisfies certain assumptions before we perform certain tests, we can make erroneous conclusions.
What assumptions do we need to make to be able to construct a confidence interval for a population mean u?
-SRS: data came from an SRS from the population of interest
-Independence: individual observations are independent; when sampling without replacement, population size N >= 10n
-Normality: the sampling distribution of x-bar is approximately Normal, population is normally distributed; Or, large sample size (n>=40)
In general, what is meant by the standard error of a statistic?
When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic.
What is the standard deviation of the sample mean x-bar?
See notes number 2.
What is the standard error of the sample mean x-bar?
See notes number 3.
Describe the similarities between a standard normal distribution and a t distribution.
The density curves of the t distributions are similar in shape to the standard normal curve. They are symmetric about zero, single peaked, and bell-shaped.
Describe the differences between a standard normal distribution and a t distribution.
There is a different t distribution for each sample size. We specify a particular t distribution by giving its degrees of freedom. The spread of the t distributions is a bit greater than that of the standard normal distribution. The t distributions have more probability in the tails and less in the center than does the standard normal. Therefore, the t-distribution is shorter than the normal curve.
How do you calculate the degrees of freedom for a t distribution?
If the SRS has size n, the t-statistic has the t distribution with n-1 degrees of freedom.
What happens to the t distribution as the degrees of freedom increase?
As the degrees of freedom k increase, the t curve approaches the N(0,1) curve ever more closely.
How would you construct a level C confidence interval for u if o is unknown?
See notes number 7.1
The z-Table gives the area under the standard normal curve to the left of z. What does the t-Table give?
The t-Table gives the area under the t-curve with n degrees of freedom to the left of it.
In a matched pairs t-procedure, what is u, the parameter of interest?
The parameter u in a matched pairs t-procedure is the mean difference in the responses to the two treatments within matched pairs of subjects in the entire problem.
Samples from normal distribution have very few outliers. If your data contains outliers, what does this suggest?
That the distribution is not normal.
If the size of the SRS is less than 15, when can we use t-procedures on the data?
Use t procedures if the data are close to normal. If the data are clearly nonnormal or if outliers are present, do not use t.
If the size of the SRS is at least 15, when can we use t-procedures on the data?
The t-procedures can be used except in the presence of outliers or strong skewness.
If the size of the SRS is at least 40, when can we use t-procedures on the data?
The t-procedures can be used even for clearly skewed distributions when the sample is large, roughly n>=40.
Inference for a Population Proportion
See notes problems 1-5, 7
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