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garik1379 [7]
3 years ago
10

In a sample of 8 high school students, they spent an average of 28.8 hours each week doing sports with a sample standard deviati

on of 3.2 hours. Find the 95% confidence interval, assuming the times are normally distributed.

Mathematics
1 answer:
Alja [10]3 years ago
4 0

Answer:

26.12\:

Step-by-step explanation:

Since the population standard deviation \sigma is unknown, and the sample standard deviation s, must replace it, the t distribution  must be used for the confidence interval.

The sample size is n=8.

The degree of freedom is df=n-1, \implies df=8-1=7.

With 95% confidence level, the \alpha-level(significance level) is 5%.

Hence with 7 degrees of freedom, t_{\frac{\alpha}{2} }=2.365. (Read from the t-distribution table see attachment)

The 95% confidence interval can be found by using the formula:

\bar X-t_{\frac{\alpha}{2}}(\frac{s}{\sqrt{n} } )\:.

The sample mean is \bar X=28.8 hours.

The sample sample standard deviation is s=3.2 hours.

We now substitute all these values into the formula to obtain:

28.8-2.365(\frac{3.2}{\sqrt{8} } )\:.

26.12\:

We are 95% confident that the population mean is between 26.12 and 31.48  hours.

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