Answered

A certain retail store surveyed a random sample of 50 customers. The survey asked the customers to respond to the following statement: "I received excellent customer service during my visit." (Assume the only answers are Agree or Disagree). The results are in the table below.




Agree		DisagreeAgreeDisagreeAgreeAgreeAgreeAgreeAgreeDisagree

Agree		DisagreeAgreeDisagreeAgreeDisagreeAgreeDisagreeAgreeDisagree

Agree		DisagreeAgreeDisagreeAgreeDisagreeAgreeDisagreeAgreeDisagree

Agree		DisagreeAgreeDisagreeAgreeDisagreeAgreeDisagreeAgreeDisagree

Agree		AgreeAgreeDisagreeDisagreeAgreeDisagreeAgreeAgreeDisagree






Calculate the $99\%$ confidence interval for the population proportion of those that said Agree. Round your answers to three decimal places.



$\square < p < \square$

Verified Answer :

  • The sample proportion of 'Agree' is \(\hat{p} = \frac{28}{50} = 0.56\).
  • Using a z-score of \(2.576\) for a 99% confidence level, the margin of error is \(ME \approx 0.181\).
  • The 99% confidence interval is \(\hat{p} \pm ME\), which is approximately \(0.379\) to \(0.741\).
  • Final Answer: \(\boxed{0.379 < p < 0.741}\).

Explanation

  1. Calculate the sample proportion of "Agree" responses

To find the confidence interval, we first determine the sample proportion of customers who responded "Agree". There are \(50\) total customers. By counting the "Agree" responses in the provided table:

  • Row 1: \(7\) Agree
  • Row 2: \(5\) Agree
  • Row 3: \(5\) Agree
  • Row 4: \(5\) Agree
  • Row 5: \(6\) Agree

Total "Agree" responses \(x = 7 + 5 + 5 + 5 + 6 = 28\).
The sample proportion \(\hat{p}\) is given by:
\[\hat{p} = \frac{x}{n} = \frac{28}{50} = 0.56\]

  1. Identify the critical z-score for 99% confidence

For a \(99\%\) confidence level, we look for the critical value \(z^*\) such that the area between \(-z^*\) and \(z^*\) is \(0.99\). This leaves \(0.005\) in each of the two tails of the normal distribution. Referring to standard statistical tables, the critical z-score is approximately \(z^* \approx 2.576\).

  1. Calculate the standard error and margin of error

The standard error (\(SE\)) of the sample proportion is calculated as:
\[SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.56(1 - 0.56)}{50}} = \sqrt{\frac{0.56 \times 0.44}{50}} = \sqrt{\frac{0.2464}{50}} \approx 0.0702\]
The margin of error (\(ME\)) is then:
\[ME = z^* \times SE = 2.576 \times 0.0702 \approx 0.1808\]

  1. Construct the confidence interval and final result

The confidence interval is given by \(\hat{p} \pm ME\).
Lower bound: \(0.56 - 0.1808 = 0.3792\)
Upper bound: \(0.56 + 0.1808 = 0.7408\)
Rounding these values to three decimal places, we get the interval \(\boxed{0.379 < p < 0.741}\).

Examples

Confidence intervals are widely used by businesses to gauge customer sentiment. For instance, if a store wants to know the true proportion of all customers who are satisfied with their service, they can take a small random sample and calculate a confidence interval to estimate the population proportion with a specific level of certainty, like 99%. If \(\hat{p} = 0.56\) and the margin of error is \(0.181\), they can be 99% confident that the actual satisfaction rate falls between \(0.379\) and \(0.741\). This helps managers make informed decisions without needing to survey every single customer.

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