Chapter 8.1, Problem 64E

Interpret a confidence interval: A survey organization drew a simple random sample of 625 households from a city of 100,000 households. The sample mean number of people in the 625 households was 2.30: and a 95% confidence interval for the mean number of people in the 100,000 households was $2.16<\mu <\mathrm{2.44.}$

True or false, and explain:

1. We are 95% confident that the mean number of people in the 625 households is between 2.16 and 2.44.
2. We are 95% confident that the mean number of people in the 100,000 households is between 2.16 and 2.44.
3. The probability is 0.95 that the mean number of people in the 100,000 households is between 2.16 and 2.44.
4. 95% of the households in the sample contain between 2.16 and 2.44 people.

To determine

To explain: Whether the statement “We are $95\%$ confident that the mean number of people in the $625$ households is between $2.16$ and $2.44$ .” is true or false.

Answer to Problem 64E

The given statement “We are $95\%$ confident that the mean number of people in the $625$ households is between $2.16$ and $2.44$ .” is false.

Explanation of Solution

Given:

Out of $100,000$ households in a certain city $625$ units are randomly selected and the mean number of family members is calculated. The mean was $2.30$ and a $95\%$ confidence is defined as $2.16<\mu <2.44$ .

The confidence interval is constructed for the population mean $\left(\mu \right)$ ; the mean number of family members in all $100,000$ households. Here, the mean number of members in the sample households is already calculated to $2.30$ .

Hence, the sample mean is fixed an no need to construct a confidence interval for this statistics.

Conclusion:

Therefore, we can conclude that by the confidence interval the mean of the sample is not defined.

To determine

To explain: Whether the statement “We are $95\%$ confident that the mean number of people in the $100,000$ households is between $2.16$ and $2.44$ .” is true or false.

Answer to Problem 64E

The statement “We are $95\%$ confident that the mean number of people in the $100,000$ households is between $2.16$ and $2.44$ .” is true.

Explanation of Solution

Out of $100,000$ households in a certain city $625$ units are randomly selected and the mean number of family members is calculated. The mean was $2.30$ and a $95\%$ confidence is defined as $2.16<\mu <2.44$ .

By the notation $\mu$ the population mean is denoted. Hence, the interval $2.76<\mu <2.88$ with $95\%$ confidence level is to definevariation of population mean.

The population that is discussed in this case is all $100,000$ households in the city. therefore, he population mean should be the mean number of members in the total number of households.

Conclusion:

Hence, the mean number of people in the $100,000$ households have $95\%$ confidence interval between the given interval.

Conclusion:

Here, in the statement it is said that the $95\%$ confidence interval is for the sample mean. Therefore, this statement is said to be false.

To determine

To explain: Whether the statement “The probability is $0.95$ that the mean GPA of all students in the college is between $2.16$ and $2.44$ .” is true or false.

Answer to Problem 64E

The statement “The probability is $0.95$ that the mean GPA of all students in the college is between $2.16$ and $2.44$ .” is true.

Explanation of Solution

Out of $100,000$ households in a certain city $625$ units are randomly selected and the mean number of family members is calculated. The mean was $2.30$ and a $95\%$ confidence is defined as $2.16<\mu <2.44$ .

The confidence level of an interval denotes the chance/ probability of the relevant parameter to be within the interval.

Here, the interval $2.16<\mu <2.44$ is constructed for the population mean with $95\%$ confidence level. Hence, there is $95\%$ chance that the population mean is between $2.16$ and $2.44$ .

Generally, a probability is expressed as a decimal or fraction. Hence, we should convert this percentage into a decimal to check the statement.

  $95\%=\frac{35}{100}=0.95$

Conclusion:

Therefore we can conclude that the probability for the mean number of people in total number of households is between the confidence interval is $0.95$ .

To determine

To explain: Whether the statement “ $95\%$ of the households in the sample contain between $2.16$ and $2.44$ people.” is true or false.

Answer to Problem 64E

The statement “ $95\%$ of the households in the sample contain between $2.16$ and $2.44$ people.” is false.

Explanation of Solution

Out of $100,000$ households in a certain city $625$ units are randomly selected and the mean number of family members is calculated. The mean was $2.30$ and a $95\%$ confidence is defined as $2.16<\mu <2.44$ .

As mentioned in the above parts, the confidence interval is defined for the population mean. Hence all the estimations can be made only for the mean number of people in the total number of households in the city.

In the statement the number of peoples in each household is considered individually. It says there is a $95\%$ chance for the number of people in a randomly selected household to be between $2.16$ and $2.44$ .

Conclusion:

Therefore, we can conclude that the given confidence interval does not represent a proportion of the population.