IXL - Interpret confidence intervals for population means (Precalculus practice)

Westford's chamber of commerce wants to help promote local restaurants. To start, it gave a survey to 125 randomly chosen residents about their recent restaurant visits. From the survey results, the chamber of commerce calculated a

95%

confidence interval of

(

5.4

6.2

)

for the mean number of times Westford residents ate out last month.

Is the following conclusion valid?

If the chamber of commerce conducts another survey, there is a

95%

chance that the mean number of times Westford residents ate out last month will be in the new survey's

95%

confidence interval.

yes

ref_doc_title.

Learn with an example

question

A company is thinking about offering shuttles to help its employees get to work. To aid in making its decision, the company surveyed 75 randomly chosen employees. From the survey results, the company calculated a

90%

confidence interval of

(

11.4

)

for the mean time employees at the company spend on their morning commutes.

Is the following conclusion valid?

If 100 more surveys are conducted (each using a sample with members chosen randomly and independently), it is expected that exactly 90 of them will each produce a

90%

confidence interval that contains its sample mean.

yes

key idea

You can use a random sample to calculate a

confidence interval for the population mean. The population mean is the mean of all data values in a population.

confidence interval depends on the sample it is calculated from. Different samples can yield different

confidence intervals.

The center of any confidence interval is the sample mean. The sample mean is the mean of all data values in the sample.

A confidence interval may or may not contain the population mean. A

confidence interval will contain the population mean for

of samples whose elements were chosen randomly and independently.

solution

Look at the conclusion:

If 100 more surveys are conducted (each using a sample with members chosen randomly and independently), it is expected that exactly 90 of them will each produce a

90%

confidence interval that contains its sample mean.

The sample mean is the average time employees in the survey spend on their morning commutes.

A confidence interval for a sample always contains its sample mean. In fact, it is symmetric around the sample mean.

If 100 more samples are taken, it is guaranteed that all 100, not just 90 of them, will each produce a

90%

confidence interval that contains the sample mean.

So, the conclusion is not

valid.

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