a.a statistics practitioner randomly sampled 100 observations from a population with a standard deviation of 5 and found that x is 10. Estimate the population mean with 90% confidence.
Repeat (a) with a sample sixe of 25
To estimate the population mean with a 90% confidence interval, we can use the formula:
Confidence Interval = x ± (Z * (σ/√n))
Where:
x = sample mean
Z = Z-score for the desired level of confidence
σ = population standard deviation
n = sample size
(a) Sample Size = 100:
Given that x = 10, σ = 5, and n = 100, we first need to determine the Z-score for a 90% confidence level.
To find the Z-score, we refer to the Z-table or use statistical software. For a 90% confidence level, the Z-score is approximately 1.645.
Putting the values into the formula:
Confidence Interval = 10 ± (1.645 * (5/√100))
= 10 ± (1.645 * (5/10))
= 10 ± (1.645 * 0.5)
= 10 ± 0.8225
The 90% confidence interval for the population mean is (9.1775, 10.8225).
(b) Sample Size = 25:
Using the same formula, with x = 10, σ = 5, and n = 25, we need to calculate the Z-score for a 90% confidence level.
Again, the Z-score for a 90% confidence level is approximately 1.645.
Confidence Interval = 10 ± (1.645 * (5/√25))
= 10 ± (1.645 * (5/5))
= 10 ± (1.645 * 1)
= 10 ± 1.645
The 90% confidence interval for the population mean with a sample size of 25 is (8.355, 11.645).
To estimate the population mean with a given level of confidence, we can use the formula for confidence interval:
Confidence Interval = sample mean ± (critical value * standard deviation / square root of sample size)
(a) For a sample size of 100 and a 90% confidence level:
Given:
Sample size (n) = 100
Standard deviation (σ) = 5
Sample mean (x̄) = 10
Step 1: Determine the critical value
The critical value corresponds to the desired confidence level. For a 90% confidence level, the critical value can be found using a t-table or calculator. In this case, the critical value is approximately 1.645 for a sample size of 100.
Step 2: Calculate the confidence interval
Confidence Interval = 10 ± (1.645 * 5 / √100)
Confidence Interval = 10 ± 0.823
Therefore, the 90% confidence interval for the population mean is (9.177, 10.823).
(b) For a sample size of 25 and a 90% confidence level:
Given:
Sample size (n) = 25
Standard deviation (σ) = 5
Sample mean (x̄) = 10
Step 1: Determine the critical value
Using the t-table or calculator, the critical value for a 90% confidence level with a sample size of 25 is approximately 1.711.
Step 2: Calculate the confidence interval
Confidence Interval = 10 ± (1.711 * 5 / √25)
Confidence Interval = 10 ± 1.711
Therefore, the 90% confidence interval for the population mean with a sample size of 25 is (8.289, 11.711).
Remember, as the sample size decreases, the confidence interval becomes wider, which creates more uncertainty in the estimation of the population mean.