A popular menu item at Starbucks is the double smoked bacon cheddar and egg sandwich. The published nutrition facts indicate 260 calories from fat. Suppose a random sample of 50 sandwiches is taken and the calorie count from fat is measured. The sample mean was 273.41 calories. Suppose the population standard deviation is known as 44.8 calories. Calculate a 90% confidence interval for the true mean amount of calories in the double smoked bacon cheddar and egg sandwich.
The Correct Answer and Explanation is:
To calculate a 90 percent confidence interval for the true mean calories from fat in the Starbucks double smoked bacon cheddar and egg sandwich, we use the formula for the confidence interval when the population standard deviation is known:CI=xˉ±z∗(σn)\text{CI} = \bar{x} \pm z^* \left( \frac{\sigma}{\sqrt{n}} \right)CI=xˉ±z∗(nσ)
Where:
- xˉ=273.41\bar{x} = 273.41xˉ=273.41 is the sample mean
- σ=44.8\sigma = 44.8σ=44.8 is the population standard deviation
- n=50n = 50n=50 is the sample size
- z∗z^*z∗ is the critical z value for a 90 percent confidence level, which is approximately 1.645
Now compute the standard error:SE=44.850≈44.87.071≈6.336\text{SE} = \frac{44.8}{\sqrt{50}} \approx \frac{44.8}{7.071} \approx 6.336SE=5044.8≈7.07144.8≈6.336
Now compute the margin of error:ME=1.645×6.336≈10.42\text{ME} = 1.645 \times 6.336 \approx 10.42ME=1.645×6.336≈10.42
Finally, compute the confidence interval:273.41±10.42=(262.99,283.83)273.41 \pm 10.42 = (262.99, 283.83)273.41±10.42=(262.99,283.83)
Interpretation and Explanation
The 90 percent confidence interval for the true mean number of calories from fat in the double smoked bacon cheddar and egg sandwich is from approximately 262.99 to 283.83 calories. This means we are 90 percent confident that the actual average calorie count from fat for all such sandwiches lies within this range.
This confidence interval is based on a random sample of 50 sandwiches, where the sample mean was 273.41 calories and the population standard deviation is assumed to be known at 44.8 calories. Since the sample size is sufficiently large and the population standard deviation is known, we can apply the z-distribution for this calculation.
The margin of error, which is the range added to and subtracted from the sample mean, accounts for the natural variability expected from sample to sample. A 90 percent confidence level implies that if we were to take many such samples and construct a confidence interval from each, about 90 percent of those intervals would contain the true population mean.
This result suggests that the true average fat calorie count is likely higher than the published value of 260 calories. Although 260 is below the lower bound of the confidence interval, it is close enough that slight variation or rounding differences might explain the discrepancy. However, this could also prompt further investigation into possible differences between reported values and actual measurements.
