In a sample of 40 days taken at Town Talk Bakery, it was found that an average of 75 cakes per day were sold. From past surveys, the population standard deviation was found to be 25 cakes per day. a. Determine the standard error of the mean number of cakes sold per day. b. Determine the 90% confidence interval for the population mean number of cakes sold per day I c. If you wanted a wider interval, would you increase or decrease the confidence interval?
The Correct Answer and Explanation is :
Answers:
(a) Standard Error of the Mean (SEM):
[
SEM = \frac{\sigma}{\sqrt{n}} = \frac{25}{\sqrt{40}} = 3.95
]
The standard error of the mean is 3.95 cakes per day.
(b) 90% Confidence Interval for the Population Mean:
Using the Z-score for a 90% confidence level (which is 1.645 for a two-tailed test):
[
ME = Z \times SEM = 1.645 \times 3.95 = 6.50
]
[
CI = \left( 75 – 6.50, 75 + 6.50 \right) = (68.50, 81.50)
]
Thus, the 90% confidence interval for the population mean is (68.50, 81.50) cakes per day.
(c) Would increasing or decreasing the confidence level result in a wider interval?
A wider confidence interval means there is more certainty that the true population mean is within the interval. To achieve this, we must increase the confidence level.
For example, moving from a 90% confidence interval to a 95% confidence interval would result in a larger Z-score (about 1.96 instead of 1.645), which increases the margin of error. This means the range of values in which the population mean is likely to fall is wider.
On the other hand, if we decrease the confidence level (e.g., to 80%), the Z-score would be lower, making the confidence interval narrower.
Thus, to get a wider confidence interval, we should increase the confidence level.
Now, I’ll generate an image representing this confidence interval.
Here is a visual representation of the 90% confidence interval for the population mean number of cakes sold per day.
- The blue dashed line represents the sample mean (75 cakes per day).
- The red dashed lines indicate the lower bound (68.50) and upper bound (81.50) of the confidence interval.
- The shaded red region shows the 90% confidence interval, meaning we are 90% confident that the true population mean falls within this range.
- The black curve represents the sampling distribution of the sample mean, centered at 75 with a standard error of 3.95.
This visualization helps in understanding how confidence intervals work in statistics.
