For n = 6, from N(μ, σ2) where the variance is only estimated, consider:  = .91. What is the value of q?
• For n = 6, from N(μ, σ2) where the variance is only estimated, consider: = .91. What is the value of q?
• For n = 6, from N(μ, σ2) where the variance is only estimated, consider: . Is this probability less than, greater than, or equal to .92? Explain.
• For n = 6, from N(μ, σ2) where the variance is only estimated, consider: . Is this probability less than, greater than, or equal to .95? Explain.
• Given a sample of size n from N(μ, σ2), what is ?
• Given a sample of size n from N(μ, σ2), what is (approximately)?
• If you were constructing a confidence interval for a collection of 17 data points (where the variance is unknown), what distribution exactly would you use?
• Suppose you have decided to gather some known fixed number n of observations, and create a confidence interval at some given confidence level c, where the variance of the underlying population will be estimated. Before you gather your n numbers, don’t know how big your margin of error will be. Explain why.
• Suppose you have decided to gather some known fixed number n of observations, and create a confidence interval at some given confidence level c, where the variance of the underlying population will be estimated. Before you gather your n numbers, you don’t know how long your confidence interval will be. Explain why.
• Calculate the appropriate ratio to determine how many times larger the length of a confidence interval (at the 96% confidence level) is when the variance is only estimated from a data set with 6 observations (and a mean of 4 and standard deviation of 3), than when the variance is assumed to be known.
• Calculate the appropriate ratio to determine how many times larger the margin of error (at the 96% confidence level) is when the variance is only estimated from a data set with 6 observations (and a mean of 4 and standard deviation of 3), than when the variance is assumed to be known.

For n = 6, from N(μ, σ2) where the variance is only estimated, consider:  = .91. What is the value of q?

• Calculate the appropriate ratio to determine how many times larger the quantile used in a 98% confidence interval is when the variance is only estimated from a data set with 6 observations (and a mean of 4 and standard deviation of 3), than when the variance is assumed to be known.

• Calculate a 95% confidence interval for the data set {3, 5, 2, 1, 3}. Recognize that your estimate of the variance is only an estimate.
• Calculate a 99% confidence interval for a data set of 14 observations, whose mean is 6.2, and whose standard deviation is estimated to be 7.
• Calculate a 99% confidence interval for a data set of 14 observations, whose mean is 6.2, and whose variance is estimated to be 7.
• Using confidence intervals, explain how having a fairly large size of n can help you get a better sense of what the true value of  really is (since your point estimate is unlikely to be exactly correct).

• Does the size of n affect how likely it is that you will obtain a confidence interval that contains ? Explain.
130. If you are using student’s T distribution with 11 degrees of freedom, how large was your sample?

131. When the variance is unknown, what distribution do we refer to when constructing a confidence interval for the mean, when the sample has 18 observations in it?
132. What is the formula for the margin of error when the variance is unknown? Label the parts
133. What is the formula for the upper bound of a confidence interval when the variance is unknown? Label the parts – do not simply specify the margin of error; give that in terms of its parts and label them.
134. What is the formula for the lower bound of a confidence interval when the variance is unknown? Label the parts – do not simply specify the margin of error; give that in terms of its parts and label them.
• “My 95% confidence interval is (20, 30), so there is a 95% chance that the true mean μ is somewhere between 20 and 30.” Explain what is wrong with this claim.
135. A report gives a confidence interval of (223.83, 325.76). What are and the margin of error?
136. A report gives a point estimate of 156.78 with a margin of error of 23.42. What is the corresponding confidence interval?

• When investigating a probability distribution of some population of interest, we frequently explore its parameters by adverting to a different probability distribution. What is this other distribution, and why do we use it?
130. When estimating the mean of a probability distribution of some population of interest (variance known), a confidence interval is typically constructed using a normal distribution, even if the underlying population distribution is not normal. Why is this done?
131. If the length of my confidence interval is 9, what is my margin of error?
132. If my margin of error is 5, what is the length of my confidence interval?

133. If my margin of error is 5, how long is my confidence interval if I double the size of my data (assume all other quantities remain the same)?
134. If my margin of error is 5, what is it if I double the size of my data (assume all other quantities remain the same)?
135. If the length of my confidence interval is 8, how long is it if I double the size of my data (assume all other quantities remain the same)?
136. Construct a margin of error (at the 94% confidence level) for the mean for the data set {3, 2, -2, 0}.
• Suppose q and F are the quantile function and cdf of a given distribution. What is F(q(p))? Explain.

• Suppose q and F are the quantile function and cdf of a given distribution. What is q(F(x))? Explain.
130. Describe the role that the Central Limit Theorem plays in our construction of confidence intervals.
131. If you have two data sets that yield identical estimates of the variance, but for the first one you used 20 observations and for the second you used 30, which one will have the larger margin of error? Calculate/explain/prove your answer.
• Suppose you have decided to gather some fixed number n of observations, and create a confidence interval at some confidence level c, where the variance of the underlying population is known to be 2. Before you even gather your n numbers, you already know how big your margin of error will be. Explain why.
• Suppose you have decided to gather some fixed number n of observations, and create a confidence interval at some confidence level c, where the variance of the underlying population is known to be 2. Before you even gather your n numbers, you already know how long the confidence interval will be. Explain why.
132. In estimating the mean of X (where X has a known standard deviation of 5), your 94% confidence interval had a total length of (approximately) 7. What is the length of the confidence interval if you change the confidence level to 98%?

133. In estimating the mean of X (where X has a known standard deviation of 5), the size of your 94% margin of error was (approximately) 7. What is the size of the margin of error if you change the confidence level to 98%?
134. In estimating the mean of X, your 94% confidence interval had a total length of (approximately) 7. What is the (approximate) size of n if the known standard deviation of X is 3?

Using question 131)
135. In estimating the mean of X, the size of your 94% margin of error was (approximately) 7. What is the (approximate) size of n if the known standard deviation of X is 3?
136. In a given study, how (if at all) would increasing the size of the error probability  affect the length of a confidence interval?