Q 8.76 - Elementary Statistics

Question

Poverty and Dietary Calcium. Refer to Exercise 8.70.

a. Find a $90\%$ confidence interval for $\mu$.

b. Why is the confidence interval you found in part (a) shorter than the one in Exercise 8.70?

c. Draw a graph similar to that shown in Fig $8.6$ on page 326 to display both confidence intervals.

d. Which confidence interval yields a more accurate estimate of $\mu$ ? Explain your answer.

Step-by-Step Solution

Answer

a. The required confidence level is $(874.502,1,020.276)$

b. Width of interval is smaller if confidence level decrease.

c. 

d. 

Step 1: Given Information

Confidence level given is $90\%, 95\%$

Step 2: Determining the 90 % confidence level for μ

Follow following steps in MINTAB:

• Click Stat > Basic Statistics > 1-Sample C
• In Samples of column, click CALCIUM column.
• Enter $180$ in standard deviation.
• Go through Options, enter Confidence level as $80$.
• In alternative, choose not equal.
• Press OK in all dialogue boxes.

From output of MINTAB, $90\%$ confidence interval for $\mu$ is $(874.502,1,020.276)$

Step 3: Explanation of why confidence level is shorter

It is due to the fact that $95\%$ confidence level is used in Exercise 8.70.

In above part, $90\%$ confidence level is used.

Also, width of interval is smaller if confidence level decrease. 

Step 4: Graph for both cases

For $90\%$ confidence level of $\mu$

For $95\%$ confidence level of $\mu$

Step 5: Explanation of which confidence level yields more accurate estimate of μ

Smaller confidence interval provide more accurate estimation. Hence, $90\%$ confidence level provides more accurate estimation of $\mu$.