8.2 A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case

Use the following information to answer the next five exercises. A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hours with a sample standard deviation of 0.5 hours.

1.

Identify the following:

Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.6.

Identify the following:

Use the following information to answer the next 13 exercises: The data in Table \(\PageIndex{2}\) are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let \(X\) = the number of colors on a national flag.

Use the following information to answer the next two exercises: Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions.25.

When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 90% confident that the population proportion is estimated to within 0.05?

If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey? Why?

27.

Identify the following:

Use the following information to answer the next five exercises: Of 1,050 randomly selected adults, 360 identified themselves as manual laborers, 280 identified themselves as non-manual wage earners, 250 identified themselves as mid-level managers, and 160 identified themselves as executives. In the survey, 82% of manual laborers preferred trucks, 62% of non-manual wage earners preferred trucks, 54% of mid-level managers preferred trucks, and 26% of executives preferred trucks.32.

We are interested in finding the 95% confidence interval for the percent of executives who prefer trucks. Define random variables \(X\) and \(p^{\prime}\) in words.

33.

Which distribution should you use for this problem?

Construct a 90% confidence interval, and state the confidence interval and the error bound.

41.

What would happen to the confidence interval if the level of confidence were 95%?

Use the following information to answer the next 16 exercises: The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population.

Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds.58.

Identify the following:

Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal.63.

Identify the following:

Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds.70.

Identify the following:

Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let \(X\) = the age of a Winter Foothill College student.80.

In one complete sentence, explain what the interval means.

93.

Using the same mean, standard deviation, and level of confidence, suppose that \(n\) were 69 instead of 25. Would the error bound become larger or smaller? How do you know?

Using the same mean, standard deviation, and sample size, how would the error bound change if the confidence level were reduced to 90%? Why?

95.

Find the value of the sample size needed to if the confidence interval is 90% that the sample proportion and the population proportion are within 4% of each other. The sample proportion is 0.60. Note: Round all fractions up for \(n\).

Find the value of the sample size needed to if the confidence interval is 95% that the sample proportion and the population proportion are within 2% of each other. The sample proportion is 0.650. Note: Round all fractions up for \(n\).

97.

Find the value of the sample size needed to if the confidence interval is 96% that the sample proportion and the population proportion are within 5% of each other. The sample proportion is 0.70. Note: Round all fractions up for \(n\).

Find the value of the sample size needed to if the confidence interval is 90% that the sample proportion and the population proportion are within 1% of each other. The sample proportion is 0.50. Note: Round all fractions up for \(n\).

99.

Find the value of the sample size needed to if the confidence interval is 94% that the sample proportion and the population proportion are within 2% of each other. The sample proportion is 0.65. Note: Round all fractions up for \(n\).

Find the value of the sample size needed to if the confidence interval is 95% that the sample proportion and the population proportion are within 4% of each other. The sample proportion is 0.45. Note: Round all fractions up for \(n\).

101.

Find the value of the sample size needed to if the confidence interval is 90% that the sample proportion and the population proportion are within 2% of each other. The sample proportion is 0.3. Note: Round all fractions up for \(n\).