Question

You measure 41 textbooks' weights, and find they have a mean weight of 78 ounces. Assume the population standard deviation is 12 ounces. Based on this, construct a 95% confidence interval for the true population mean textbook weight.Give your answers as decimals, to two places < μ <

Answer 1

To construct the confidence interval, we first need to determine which distribution we need to use. Since the sample size is 41 (this is the number of books) and the standard deviation of the population is known we will use the normal distribution to construct our interval, that is, the interval is given by:

`\bar{x}\pm z\sigma_{\bar{x}}`

where:

`\begin{gathered} \bar{x}\text{ is the mean of the sample} \\ \sigma_{\bar{x}}=(\sigma)/(√(n))\text{ is the standard deviation of the mean} \\ z\text{ is the z-value for the given confidence interval } \end{gathered}`

We know that the point estimate of the mean is 78 ounces. The standard deviation of the mean is:

`\sigma_{\bar{x}}=(12)/(√(41))=1.874`

The z-value for a 95% confidence interval is 1.96, then we have:

`\begin{gathered} \bar{x}-z\sigma_{\bar{x}}=78-(1.96)(1.874)=74.33 \\ \bar{x}+z\sigma_{\bar{x}}=78+(1.96)(1.874)=81.68 \end{gathered}`

Therefore, the confidence interval of the mean is:

`74.33<\mu<81.68`