Question

Recently, a case of food poisoning was traced to a particular restaurant chain. The source was identified and corrective actions were taken to make sure that the food poisoning would not reoccur. Despite the response from the restaurant chain, many consumers refused to visit the restaurant for some time after the event. A survey was conducted three months after the food poisoning occurred, with a sample of 319 former customers contacted. Of the 319 contacted, 29 indicated that they would not go back to the restaurant because of the potential for food poisoning. Construct a 95 percent confidence interval for the true proportion of the market who still refuse to visit any of the restaurants in the chain three months after the event.

Answer 1

To construct a 95 percent confidence interval for the true proportion of the market who still refuse to visit any of the restaurants in the chain three months after the event, we use the formula CI = p-hat ± z * sqrt((p-hat * (1 - p-hat)) / n). Substituting the given values, we find that the 95% confidence interval is approximately 0.082 to 0.10.

Step-by-step explanation:

To construct a 95 percent confidence interval for the true proportion of the market who still refuse to visit any of the restaurants in the chain three months after the food poisoning incident, we can use the formula:

CI = p-hat ± z * sqrt((p-hat * (1 - p-hat)) / n)

Where:

• CI is the confidence interval
• p-hat is the sample proportion
• z is the z-score corresponding to the desired confidence level (for 95% confidence, z = 1.96)
• n is the sample size

In this case, the sample proportion is 29/319 = 0.091, the z-score for a 95% confidence level is 1.96, and the sample size is 319.

Substituting these values into the formula:

CI = 0.091 ± 1.96 * sqrt((0.091 * (1 - 0.091)) / 319)

Simplifying the expression:

CI = 0.091 ± 1.96 * sqrt(0.0828 / 319)

CI = 0.091 ± 1.96 * 0.0045

CI = 0.091 ± 0.0088

Therefore, the 95% confidence interval for the true proportion of the market who still refuse to visit any of the restaurants in the chain three months after the event is approximately 0.082 to 0.10.