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

The following random sample was selected from a normal population: 9, 11, 8, 10, 14, 8. Construct a 95% confidence interval for the population mean $\mu .$

What is $z _ { \alpha / 2 }$ when $\alpha = 0.06$ ?

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 99% 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 $12,760 with a standard deviation of $700. Suppose that the interval is Calculated to be ($12,264.09, $13,255.91). How could we alter the sample size and the confidence Coefficient in order to guarantee a decrease in the width of the interval?

Suppose you selected a random sample of n = 7 measurements from a normal distribution. Compare the standard normal z value with the corresponding t value for a 90% confidence interval.

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 99% confidence interval for The mean was calculated to be ($2,181,260, $5,836,180). Explain what the phrase ʺ99% confidentʺ Means.

Suppose you selected a random sample of n = 29 measurements from a normal distribution. Compare the standard normal z value with the corresponding t value for a 95% confidence interval.

Let $t _ { 0 }$ be a specific value of $t$ . Find $t _ { 0 }$ such that the following statement is true: $P \left( t \leq t _ { 0 } \right) = .005$ where $\mathrm { df } = 20$ .

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.

What type of car is more popular among college students, American or foreign? One hundred fifty-nine college students were randomly sampled and each was asked which type of car he or She prefers. A computer package was used to generate the printout below of a 99% confidence Interval for the proportion of college students who prefer American automobiles. SAMPLE PROPORTION $= .396$ SAMPLE SIZE $= 159$ UPPER LIMIT $= .496$ LOWER LIMIT $= .296$ Which of the following is a correct practical interpretation of the interval?

The mean replacement time for a random sample of 12 CD players is $8.6$ years with a standard deviation of $4.4$ years. Construct the $99 \%$ confidence interval for the population variance, $\sigma ^ { 2 }$ . Assume the data are normally distributed, and round to the nearest hundredth when necessary.

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 $12,900 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.

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

A 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($100,951, $115,349). To make more useful inferences from the data, it is desired to reduce the Width of the confidence interval. Which of the following will result in a reduced interval width?

In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a __________ interval.

A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 330 college students produced the following 95% confidence interval for the Proportion of college students who prefer one of the colas: (.330, .469). What additional Assumptions are necessary for the interval to be valid?

A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60-year-old male non-smoker. He randomly sampled 65 subjects (60-year-old male non-smokers)and constructed the following 95 percent confidence interval for The mean cost of the term life insurance: ($850.00, $1050.00). What value of alpha was used to Create this confidence interval?

For the given combination of $\alpha$ and degrees of freedom (df), find the value of $\chi _ { ( 1 - \alpha / 2 ) } ^ { 2 }$ that would be used to find the upper endpoint of a confidence interval for $\sigma ^ { 2 }$ . $\alpha = 0.01 , \mathrm { df } = 30$

Find the value of $t _ { 0 }$ such that the following statement is true: $P \left( - t _ { 0 } \leq t \leq t _ { 0 } \right) = .95$ where df $= 15$ .

The director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 49 different 24-hour periods and determines the number of admissions for each. For this sample, $\bar { x } = 15.3$ and $s ^ { 2 } = 25$ . Estimate the mean number of admissions per 24 -hour period with a $95 \%$ confidence interval.

A 90% confidence interval for the mean percentage of airline reservations being canceled on the day of the flight is (1.5%, 4.2%). What is the point estimator of the mean percentage of reservations That are canceled on the day of the flight?