https://file.aiquickdraw.com/mathbot/file/1730586380194wxrbopcd.jpg

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.

The website of the large corporation CorpPlus claims that 93% of the population of all employees say the corporation is a great place to work. You work for a competitor and think that this percentage is not a true indicator of CorpPlus's employee satisfaction. To test the claim from the website, you will survey a random sample of 55 employees from CorpPlus, and ask them whether they think the corporation is a great place to work. Follow the steps below to construct a 90% confidence interval for the population proportion of all employees who say the corporation is a great place to work. Then state whether the confidence interval you construct contradicts the website's claim.

Learn how to construct a 90% confidence interval for the population proportion of employees who think CorpPlus is a great place to work. This problem includes concepts like sampling distributions, standard error calculation, and margin of error for confidence intervals. We analyze whether the sample data supports the company's claim of 93% satisfaction.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation