Questions: A fast food restaurant executive wishes to know how many fast food meals adults eat each week. They want to construct a 99% confidence interval for the mean and are assuming that the population standard deviation for the number of fast food meals consumed each week is 1.5. The study found that for a sample of 361 adults the mean number of fast food meals consumed per week is 7.4. Construct the desired confidence interval. Round your answers to one decimal place.

A fast food restaurant executive wishes to know how many fast food meals adults eat each week. They want to construct a 99% confidence interval for the mean and are assuming that the population standard deviation for the number of fast food meals consumed each week is 1.5. The study found that for a sample of 361 adults the mean number of fast food meals consumed per week is 7.4. Construct the desired confidence interval. Round your answers to one decimal place.

Transcript text: A fast food restaurant executive wishes to know how many fast food meals adults eat each week. They want to construct a $99 \%$ confidence interval for the mean and are assuming that the population standard deviation for the number of fast food meals consumed each week is 1.5. The study found that for a sample of 361 adults the mean number of fast food meals consumed per week is 7.4. Construct the desired confidence interval. Round your answers to one decimal place.

Solution

Solution Steps

Step 1: Given Information

We are tasked with constructing a $99\%$ confidence interval for the mean number of fast food meals consumed per week by adults. The following information is provided:

Sample mean ($\bar{x}$): $7.4$
Sample size ($n$): $3611$
Population standard deviation ($\sigma$): $1.5$
Confidence level: $99\%$

Step 2: Calculate the Z-Score

For a $99\%$ confidence level, the corresponding Z-score can be found using standard normal distribution tables or calculators. The Z-score for $99\%$ confidence is approximately $z = 2.576$.

Step 3: Calculate the Margin of Error

The margin of error ($E$) is calculated using the formula: \[ E = z \cdot \frac{\sigma}{\sqrt{n}} \] Substituting the known values: \[ E = 2.576 \cdot \frac{1.5}{\sqrt{3611}} \approx 2.576 \cdot 0.025 \approx 0.0644 \]

Step 4: Construct the Confidence Interval

The confidence interval is given by: \[ \bar{x} \pm E \] Calculating the lower and upper endpoints: \[ \text{Lower endpoint} = 7.4 - 0.0644 \approx 7.3356 \quad \text{(rounded to 7.3)} \] \[ \text{Upper endpoint} = 7.4 + 0.0644 \approx 7.4644 \quad \text{(rounded to 7.5)} \]

Step 5: Final Confidence Interval

Thus, the $99\%$ confidence interval for the mean number of fast food meals consumed per week is: \[ (7.3, 7.5) \]