The mean of a sample of 25 was calculated as xbar = 500. The sample was randomly drawn from a population whose standard deviation is 15. Estimate the population mean with 99% confidence.
B.) Repeat part (a) changing the population standard deviation to 30.
C.) Repeat part (a) changing the population standard deviation to 60.
D.) Describe what happens to the confidence interval estimate when the standard deviation is increased.
To estimate the population mean with 99% confidence, we can use the formula for a confidence interval:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Where:
- Sample Mean (x̄) is the mean of the sample, given as 500.
- Critical Value is the value corresponding to the desired confidence level and sample size. We can find this value using a t-distribution table or a statistical calculator.
- Standard Error is the standard deviation of the sample divided by the square root of the sample size.
a) Since the sample size is 25 and we want a 99% confidence interval, we need to find the critical value for a 99% confidence level with 24 degrees of freedom (25 - 1).
Using a t-distribution table or calculator, we find that the critical value for a 99% confidence level with 24 degrees of freedom is approximately 2.797.
Now, we can calculate the standard error:
Standard Error = Population Standard Deviation / √Sample Size
For part (a), when the population standard deviation is 15:
Standard Error = 15 / √25 = 3
Substituting the values into the formula for the confidence interval:
Confidence Interval = 500 ± (2.797 * 3)
Thus, the confidence interval estimate for part (a) when the population standard deviation is 15 is (493.61, 506.39).
b) For part (b), when the population standard deviation is 30, the only value that changes is the standard error calculation. The rest of the steps remain the same.
Standard Error = 30 / √25 = 6
Substituting the new standard error into the formula:
Confidence Interval = 500 ± (2.797 * 6)
Therefore, the confidence interval estimate for part (b) when the population standard deviation is 30 is (486.21, 513.79).
c) For part (c), when the population standard deviation is 60, again, only the standard error calculation changes.
Standard Error = 60 / √25 = 12
Substituting the new standard error into the formula:
Confidence Interval = 500 ± (2.797 * 12)
Thus, the confidence interval estimate for part (c) when the population standard deviation is 60 is (475.64, 524.36).
d) When the standard deviation is increased, the confidence interval estimate becomes wider. This means that there is more uncertainty in estimating the population mean because the data points are more spread out. A larger standard deviation indicates a greater variation in the population, leading to a larger range of possible values for the true population mean.