The Police Department is investigating the need for a speed camera to be implemented on Hasty Road. A sample of 40 cars passing through that road on a particular day were randomly selected and their speeds recorded. The speed limit on Hasty Road is60 miles per hour (mph). The mean speed for the sample was calculated as42 mph. The standard deviation of the speeds for the sample was calculated as4 mph. The standard deviation for the population of cars driving on Hasty Road is unknown.

a)Select all the techniques that are commonly used to construct a confidence interval for the mean when the population standard deviation () is unknown:

Approximate the population standard deviation () with the sample standard deviation (s)

Replace the sample size (n) withn-1

Approximate the standard normal distribution with the Student's t distribution

Decrease the confidence level to compensate for the increased margin of error

b)Calculate the upper and lower bounds of the 99% confidence interval for the mean speed traveled on Hasty Road. You may find thisStudent's t distribution tableuseful. Give your answers in mph to 2 decimal places.

Upper bound =mph

Lower bound =mph

A newspaper runs a competition each week. To enter this competition, readers must correctly complete the weekly crossword puzzle and send it in to the paper, along with a proof of purchase. All such entries go into a weekly draw for a prize. The newspaper would like to estimate the average number of entries each week. A random sample of 26 weeks are chosen, and it is observed that the average number of entries over these weeks is 816.0. The sample standard deviation is 38.2.

Calculate the 95% confidence interval for the mean number of entries in the newspaper competition. You may find thisStudent's t distribution tableuseful. Give your answers to 2 decimal places.

Tyler and Jack are both studying a numerical variable X. Both students want to estimate the population mean of this variable, and they each intend to collect a sample, calculate a sample mean and construct a 95% confidence interval. Each student will collect their own sample, but both samples will have 30 items in them.

When constructing his confidence interval, Tyler assumes a value for the population standard deviation, . In contrast, Jack does not assume a value for the population standard deviation. Jack calculates the sample standard deviation, s, and uses this in his confidence interval. However as it turns out, the sample standard deviation that Jack calculates turns out to be equal to the value assumed by Tyler for the population standard deviation.

Based on this information, it is true to say that:

both confidence intervals will be of the same width

Jack's confidence interval will be wider than Tyler's

it is impossible to tell which confidence interval will be wider

Tyler's confidence interval will be wider than Jack's