Let's go through this problem step-by-step.

Step-by-Step Solution

Step 1: Determine the values

From the table, we have:

• Sample size (n) = 55
• Number who say "Yes" = 44, so sample proportion (p-hat) = $\frac{44}{55} = 0.8$

Step 2: Point Estimate

The point estimate for the sample proportion is $p = 0.8$.

Step 3: Choose the Critical Value for a 90% Confidence Interval

For a 90% confidence interval, the z-score (critical value) is 1.645.

Step 4: Calculate the Standard Error

The formula for the standard error of a proportion is: $\text{Standard Error} = \sqrt{\frac{p \cdot (1 - p)}{n}}$ Plugging in the values: $\text{Standard Error} = \sqrt{\frac{0.8 \cdot (1 - 0.8)}{55}} = \sqrt{\frac{0.8 \cdot 0.2}{55}} = \sqrt{\frac{0.16}{55}} \approx 0.0537$

Step 5: Calculate the Margin of Error

The margin of error (ME) is calculated as: $\text{Margin of Error} = z \times \text{Standard Error} = 1.645 \times 0.0537 \approx 0.0883$

Step 6: Calculate the 90% Confidence Interval

The confidence interval is: $\text{Confidence Interval} = p \pm \text{Margin of Error} = 0.8 \pm 0.0883$ So the interval is approximately: $(0.7117, 0.8883)$

Step 7: Answer Part (c)

The claim from the website is that 93% (or 0.93) of employees think the corporation is a great place to work. Since 0.93 is outside the 90% confidence interval of (0.7117, 0.8883), this contradicts the website’s claim.

So, the correct answer to part (c) would be:

• Yes, the confidence interval contradicts the claim. The proportion 0.93 from the website is outside the 90% confidence interval.

Would you like further details on any part of this solution, or do you have any other questions?

Related Questions

1. How would the confidence interval change if we used a 95% confidence level instead of 90%?
2. What would happen to the confidence interval if the sample size were increased?
3. How does the margin of error impact the width of the confidence interval?
4. Why do we use a critical value from the z-distribution for large sample sizes?
5. How can this confidence interval help in making business decisions?

Tip

Always double-check the critical value when constructing confidence intervals, as different confidence levels have different critical z-scores.