Q 8.106. - Elementary Statistics

Question

Poverty and Dietary Calcium. Refer to Exercise $8.70$

a. Determine and interpret a $95\%$ upper confidence bound for the mean calcium intake of all people with incomes below the poverty level.

b. Compare your one-sided confidence interval in part (a) to the (twosided) confidence interval found in Exercise $8.70$

Step-by-Step Solution

Answer

Part (a) The mean calcium intake of all people with earnings below the poverty level is less than $1,020.2933$ mg per day, according to $95\%$ confidence.

Part (b) Because the $z$ value for one side with a $95\%$ confidence level is $1.645$, whereas the $z$ value for both sides with a $95\%$ confidence level is $1.96$

Step 1: Given information

From Exercise 8.70, $\overline{x}=947.4, n=18, \sigma =188$

Step 2: Concept

The formula used: The upper confidence bound $\overline{x}+z_{\frac{a}{2}}\left(\frac{\sigma }{\sqrt{n}}\right)$

Step 3: Calculation

Calculate the $95\%$ upper confidence bound for all people with incomes below the poverty level's mean calcium consumption.

From Exercise 8.70, $\overline{x}=947.4, n=18, \sigma =188$

The needed value of $z_{\alpha }$ with a $95\%$ confidence level is $1.645$, according to "Table II Areas under the standard normal curve."

Thus, the upper confidence bound is,

$$\begin{gathered} \overline{x}+z_{\frac{a}{2}}\left(\frac{\sigma }{\sqrt{n}}\right)=947.4+1.645\left(\frac{188}{\sqrt{18}}\right) \\ =947.4+72.8933 \\ =1,020.2933 \end{gathered}$$

As a result, the $95\%$upper confidence bound for the mean calcium intake of all people living in poverty is $1,020.2933$

Step 4: Explanation

Because the $z$ value for one side with a $95\%$ confidence level is $1.645$ and the $z$ value for both sides with a $95\%$ confidence level is $1.96$, it is evident that the upper confidence bound is smaller than the upper confidence limit.