Exam 7: Inferences Based on a Single Sample: Estimation With Confidence Intervals

The Central Limit Theorem guarantees an approximately normal sampling distribution for the sample mean for large sample sizes, so no knowledge about the distribution of the population is necessary for the corresponding interval to be valid.

The sampling distribution for $\hat { p }$ is approximately normal for a large sample size n, where n is considered large if both n $\hat { p }$ ? 15 and n(1 - $\hat { p }$ ) ? 15.

Suppose a large labor union wishes to estimate the mean number of hours per month a union member is absent from work. The union decides to sample 406 of its members at random and monitor the working time of each of them for 1 month. At the end of the month, the total number of hours absent from work is recorded for each employee. If the mean and standard deviation of the sample are $\bar { x } = 8.1$ hours and $s = 2.3$ hours, find a $99 \%$ confidence interval for the true mean number of hours a union member is absent per month. Rotund to the nearest thousandth. A) $8.1 \pm .294$ B) $8.1 \pm .015$ C) $8.1 \pm .194$ D) $8.1 \pm .113$

You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 95% confidence interval. You manage to obtain data on 17 recently resold 5-year-old foreign sedans of the same model. These 17 cars were resold at an average price of $12,700 with a standard deviation of $700. Create a 95% confidence interval for the true mean resale value of a 5-year-old car of that model.

Sales of a new line of athletic footwear are crucial to the success of a newly formed company. The company wishes to estimate the average weekly sales of the new footwear to within $150 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1650. How many weeks of data must be sampled for the company to get the information it desires?

A random sample of 250 students at a university finds that these students take a mean of 15.4 credit hours per quarter with a standard deviation of 2.2 credit hours. The 99% confidence interval for the mean is 15.4 ± 0.358. Interpret the interval.

The mean replacement time for a random sample of 12 CD players is 8.6 years with a standard deviation of 5.8 years. Construct the 98% confidence interval for the population variance, σ2. Assume the data are normally distributed, and round to the nearest hundredth when necessary.

A random sample of n = 100 measurements was selected from a population with unknown mean μ and standard deviation σ. Calculate a 95% confidence interval if $\bar { x } = 26 \text { and } s ^ { 2 } = 16$

What is $z \alpha / 2 \text { when } \alpha = 0.05$ ?

The confidence coefficient is the relative frequency with which the interval estimator encloses the population parameter when the estimator is used repeatedly a very large number of times.

Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval. Variable Lo 95\% CI Mean Up 95\% CI SD Parking Time 338 9.1944 10.466 11.738 11.885 Explain what the phrase "95% confident" means when working with a 95% confidence interval.

A study was conducted to determine what proportion of all college students considered themselves as full-time students. A random sample of 300 college students was selected and 210 of the students responded that they considered themselves full-time students. Which of the following would represent the target parameter of interest?

Let t0 be a particular value of t. Find a value of $t _ { 0 } \text { such that } P \left( t \leq t _ { 0 } \text { or } t \geq t _ { 0 } \right) = .1 \text { where } d f = 14 \text {. }$

When the sample size is small, confidence intervals for a population proportion are more reliable when the population proportion p is near 0 or 1.

For the given combination of $\alpha$ and degrees of freedom (df), find the value of $\chi^2 _ { ( 1 - \alpha / 2 ) }$ that would be used to find the upper endpoint of a confidence interval for $\sigma ^ { 2 }$ . $\alpha = 0.05 \text {, df } = 25$ A) $13.1197$ B) $14.6114$ C) $12.4011$ D) $40.6465$

A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 95% confidence interval was calculated to be ($2,181,260, $5,836,180). Based on the interval above, do you believe the average total compensation of CEOs in the service industry is more than $1,500,000?

A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 99% confident that the margin of error of their estimate is about 3%?

The standard deviation of a population is estimated to be 345 units. To estimate the population mean to within 50 units with 98% reliability, what size sample should be selected?

A random sample of 4000 U.S. citizens yielded 2280 who are in favor of gun control legislation. Find the point estimate for estimating the proportion of all Americans who are in favor of gun control legislation.

Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval. Variable Lo 95\% CI Mean Up 95\% CI SD Parking Time 338 9.1944 10.466 11.738 11.885 Give a practical interpretation for the 95% confidence interval given above.