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1) If you eat in at a fast-food restaurant, most of the soda machines are self-s...

Mar 15, 2024

1) If you eat in at a fast-food restaurant, most of the soda machines are self-serving. If you finish your drink, you can go back and fill up your cup as many times as you want. A local fast-food restaurant manager is concerned that people are taking advantage of filling up their drink and that the restaurant is losing money as a result. He selected a random sample of 90 customers who got a drink and are eating in the restaurant. He found that 19 of those customers are filling up more than 3 times. (a) Construct and interpret a 95 percent confidence interval for the proportion of all customers who, when ordering a drink and eating in the restaurant, will fill up their cup more than 3 times.

Solution by Steps

step 1

Identify the sample proportion (

\hat{p}

) and the sample size (

n

)

step 2

Calculate the sample proportion:

\hat{p} = \frac{\text{number of successes}}{\text{sample size}} = \frac{19}{90}

step 3

Calculate the standard error (SE) for the proportion using the formula:

SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}

step 4

Find the z-score for a 95% confidence interval, which is approximately 1.96 for a two-tailed test

step 5

Calculate the margin of error (ME) using the formula:

ME = z \times SE

step 6

Construct the confidence interval using the formula:

\text{CI} = \hat{p} \pm ME

step 7

Calculate the confidence interval:

\text{CI} = 0.2111 \pm 1.96 \times \sqrt{\frac{0.2111(1 - 0.2111)}{90}}

step 8

Interpret the confidence interval: The 95% confidence interval suggests that if we were to take many samples, 95% of them would contain the true proportion of customers who fill up their cup more than 3 times within this interval

Answer

The 95% confidence interval for the proportion of all customers who will fill up their cup more than 3 times is approximately \( 0.2111 \pm 0.0957 \), or \( (0.1154, 0.3068) \).

Key Concept

Confidence Interval for a Proportion

Explanation

A confidence interval for a proportion is a range of values that is likely to contain the population proportion with a certain level of confidence. It is calculated using the sample proportion, the z-score corresponding to the desired confidence level, and the standard error of the proportion.