A simple random sample of 60 items resulted in a sample mean of 64. The population standard deviation is 16.
a. Compute the 95% confidence interval for the population mean (to 1 decimal).
95% = mean ± 1.96 SEm
SEm = SD/√n
To compute the 95% confidence interval for the population mean, we need to use the formula:
CI = X̄ ± Z * (σ/√n)
where:
CI represents the confidence interval,
X̄ is the sample mean (64 in this case),
Z is the Z-score corresponding to the desired confidence level (95% confidence level corresponds to a Z-score of 1.96),
σ is the population standard deviation (16 in this case),
n is the sample size (60 in this case).
Let's input these values into the formula:
CI = 64 ± 1.96 * (16/√60)
Now we need to calculate the value inside the parentheses first:
16/√60 ≈ 2.065
So the formula becomes:
CI = 64 ± 1.96 * 2.065
Next, we calculate the value inside the second set of parentheses:
1.96 * 2.065 ≈ 4.05
Finally, we plug this value back into the formula:
CI = 64 ± 4.05
To find the confidence interval, we subtract and add 4.05 to the sample mean:
Lower limit = 64 - 4.05 ≈ 59.9
Upper limit = 64 + 4.05 ≈ 68.0
The 95% confidence interval for the population mean is approximately 59.9 to 68.0 (to 1 decimal).