AP Stats Ch.8&9 Verbal Exam

How are the x̅, μ, and σ used for constructing a confidence interval for a mean?

To capture the true mean μ, we use the sample mean x̅ and the standard error of the sample mean instead of the population standard deviation since its unknown to construct an interval.

How do you calculate the degrees of freedom for a t distribution?

To find the degrees of freedom, we subtract 1 from the sample size. Formula: df = n - 1

How are the p̂, P, and σ used for constructing a confidence interval for a proportion?

To infer the true proportion, we use the sample proportion to estimate the population proportion and substitute the population proportion in the standard deviation formula with the sample proportion since we don't know the population proportion.

Define a level C confidence interval.

A level C confidence interval uses sample data to estimate an unknown population parameter with an indication of how precise the estimate is and of how confident we are that the result is correct; The formula is point estimate 土 margin of error; an interval of plausible values for the parameter.

What is a Null Hypothesis?

A null hypothesis is the default belief about a parameter's value; a statement of "no difference."

Explain the difference between a one-sided alternative hypothesis and a two-sided alternative hypothesis?

A one-sided alternative hypothesis specifies a direction such as greater than or less than while a two-sided alternative hypothesis says that a parameter differs from the null hypothesis value in either direction greater than or less than.

Describe the differences between a standard normal distribution and a t distribution.

A t distribution has a larger spread than a standard normal distribution meaning it has more probability in its tails and less in the center so it's also shorter than a standard normal distribution.

What is a T-distribution? Why do we use it instead of Z when calculation significance tests for a mean?

A t distribution is a substitution for a z distribution when the population isn't Normal. We use it for inferring means because the population parameter standard deviation is unknown.

What is the value of z* for a 90% confidence interval? Include a sketch.

The z* for a confidence interval is 1.645. Refer to number 5 for a sketch.

Why do we need to be careful about multiple analyses?

We need to be careful about multiple analyses because they will probably produce some significant results by chance alone, even if all the null hypotheses are true.

How do we do a significance test for paired data?

First, we take the difference within each pair to produce a single sample, and then we follow the one-sample t procedures.

What does it mean to be 90% confident?

It means that we are 90% confident that the interval will capture the true parameter.

Sketch and label a 90% confidence interval for the standard normal curve.

couldn't upload; center: 0.90, tails: 0.05, z: 1.645

What does z* represent?

z* represents the standard Normal critical value where C% of its area is between -z* and z*; the standardized units or number of standard deviations.

In statistics, what is meant by a 95% confidence interval?

A 95% confidence interval means that we are 95% confident that the interval from __ to ___ captures the true parameter.

Explain the difference between a Type I Error and a Type II Error.

A Type I Error states that we rejected the null hypothesis when the alternative hypothesis was false while a Type II Error states that we failed to reject the null hypothesis when the null hypothesis is false.

What does a test statistic estimate?

A test statistic estimates how far a sample statistic diverges from what we would expect if the null hypothesis were true, in standardized units.

What is an alternative hypothesis?

An alternative hypothesis is the unproven belief about a parameter's value that we gather evidence for.

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 decreases. The sample size n must increase by 4 times in order to cut the margin of error in half because of the square root in the formula.

What happens to the margin of error as sigma decreases?

As sigma decreases, the margin of error decreases which results in a lower confidence level as well.

What happens to the t distribution as the degrees of freedom increase?

As the degrees of freedom increase, the t-distribution's shape resembles the normal distribution more closely.

What happens to the margin of error as z* decreases? Does this result in a higher or lower confidence level?

As z* decreases, the margin of error decreases which results in a lower confidence level.

Describe the similarities between a standard normal distribution and a t distribution.

Both distributions share similar shapes; they're both symmetrical, unimodal, and bell-shaped.

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 is z*sigma over the square of n is less than or equal to m. Solve for n.

First, multiply both sides by square root of n and then divide both sides by m. Then square both sides which solves for (z/m)2(sigma) is less than or equal to m.

How would you construct a level C confidence interval for the population mean mue if σ is unknown?

I would substitute the z* critical value for t* and σ for s which would make the formula: sample mean plus or minus t times in parentheses sample std dev over the square root of n

If a P-Value is large, what do we conclude about the Null hypothesis?

If a P-value is larger than the significance test, we fail to reject the null hypothesis because there's no convincing evidence against the null hypothesis.

If a P-Value is small, what do we conclude about the null hypothesis?

If a P-value is smaller than the significance test, we reject the null hypothesis because there's convincing evidence against the null hypothesis.

Under what assumptions is s a reasonable estimate of σ?

If the three conditions are met: Random: well-designed random sample or randomized experiment, 10%, and Normal/Large Sample.

Why is it best to have high confidence and a small margin of error?

It is best to have high confidence and a small margin of error because it provides a more precise estimate. High confidence means that the interval will almost always capture the true parameter and a small margin means the interval precisely narrows down the plausible values

What is meant by a margin of error?

Margin of error is how close we believe our estimate is, and the difference between the point estimate and the true parameter value will be less than the margin of error in C% of all samples.

What factors should be considered in using significance tests wisely?

Sample size which is affected by significance level, effect size, and power should be considered in using significance tests.

What are the conditions required to do a significance test for a mean? Show the four-step process.

State: What hypothesis do you want to test, and at what significance level? Also, define any parameters you use. Plan: Choose the appropriate inference method, and check conditions which are Random, 10%, and Normal/Large Sample which states that the population must have a Normal distribution or the sample size is greater than 30, and if the shape is unknown and the n is less than 30, use a graph of the sample data to assess the Normality of the population. Watch out for strong skewness or outliers (if present, don't use t procedures). Do: If the conditions are met, perform calculations which are: compute the test statistic and find the p-value. Conclude: Make a decision about the hypotheses in the context of the problem.

What does the four-step process for the write-up of a significance test for a proportion look like?

State: What hypothesis do you want to test, and at what significance level? Also, define any parameters you use. Plan: Choose the appropriate inference method, and check conditions. Do: If the conditions are met, perform calculations which are: compute the test statistic and find the p-value. Conclude: Make a decision about the hypotheses in the context of the problem.

Describe the four-step process for estimating a population mean

State: What parameter do you want to estimate, and at what confidence interval? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, perform the calculations: find the t* critical value, calculate the interval. Conclude: Interpret the interval in the context of the problem.

Explain the four step process for construction of a confidence interval for a population proportion.

State: What parameter do you want to estimate, and at what confidence interval? Plan: Identify the appropriate inference method. Check conditions. Do: If the conditions are met, perform the calculations: find the z* critical value, calculate the interval. Conclude: Interpret the interval in the context of the problem.

In statistics, what is meant by the P-value?

The P-value is the probability, assuming the null hypothesis is true, that the statistic would take a value as extreme as or more extreme than the one actually observed in the direction specified by the alternative hypothesis.

How small should the P-Value be in order to claim that the result is statistically significant?

The P-value should be less than the significance level.

What are the conditions required to do a significance test for a proportion?

The conditions are Random: the data come from a well-designed random sample or randomized experiment, 10%: when sampling w/o replacement, check that n ≤ (1/10)N, and Large Counts: np0 ≥ 10 and n(1-p0) where p0 is the parameter value specified by the null hypothesis

What are the conditions for constructing a confidence interval for a proportion?

The conditions are Random: well-designed random sample or randomized experiment , 10%, and the Large Counts

What are the three ways you can increase the power of a significance test? Explain how each improves the power of a significance test.

The first way is to increase the sample size which decreases the spread of both the Null and Alternative distributions, decreasing the amount of overlap between the two distributions which makes it easier to detect a difference between the null and alternative parameter values. The second way is to increase the significance level a which also increases the probability of a Type I error and decreases the probability of a Type II error, increasing the power of a test. The third way is to increase the difference between the null and alternative parameter values because it's easier to detect large differences between the null and alternative parameter values than smaller differences.

Explain how to find a level C confidence interval for an SRS of size n having unknown mean mue and known standard deviation sigma.

The formula is the point estimate or statistic plus and minus the z* critical value times in parentheses the standard deviation sigma over the square root of sample size n.

In a sampling distribution of x bar or sample means, why is the interval of numbers between x bar 土 2s called a 95% confidence interval?

The interval of numbers between x bar 土 2s is called a 95% confidence interval because 95% of the Normal distribution is within 2 standard deviations of the mean.

Where do x̅, μ, and σ come from?

The sample mean x̅ comes from the sample data, the population mean is estimated by a one-sample t interval, and the standard deviation is unknown.

Where do p̂, P, and σ come from?

The sample proportion comes from the sample data, the population proportion is estimated by the one-sample z interval, and the standard deviation is the square root of sample proportion times 1 minus the sample proportion divided by the sample size.

What is the Relationship Between the significance level a and the probability of Type I Error?

The significance level a is equal to the probability of Type I Error. The reason is when you set your significance level at a certain percent, you're willing to accept that you have a percent chance of rejecting the null hypothesis when it is not false.

What is meant by a significance level?

The significance level is the fixed value alpha that we use as a cutoff for deciding whether an observed result is too unlikely to happen by chance alone when the null hypothesis is true.

What is the standard deviation of the sample mean x bar?

The standard deviation of the sample mean x bar is the population standard deviation σ divided by the square root of sample size n.

What is the standard error of the sample mean x bar?

The standard error of the sample mean x bar is the sample standard deviation s divided by the square root of sample size n.

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 to the right.

What is the value of z* for a 95% confidence interval? Include a sketch.

The z* for a confidence interval is 1.96. Refer to number 2 for a sketch.

What is the value of z* for a 99% confidence interval? Include a sketch.

The z* for a confidence interval is 2.576.

Why might statistical significance not be the same thing as practical significance?

They might not be the same thing because the statistical significance can be a very small difference and not be practically important.

Describe how to calculate the power of a significance test.

To calculate the power of a significance test, you subtract the probability of making a Type II error, which is sometimes called beta, from 1.

How do you interpret an 80% confidence interval?

We use the sentence frame: We are 80% confident that the interval from the estimate minus the margin of error to the estimate plus the margin of error captures the true parameter and then provide context to that parameter.

In general what is meant by standard error of a statistic?

When the standard deviation of a statistic is estimated from the data, it is called the standard error of a statistic.