Given a normally distributed population, find a 95% confidence interval for the population mean;
sample mean is 119; sample std deviation is 14; sample size is 28
To find the 95% confidence interval for the population mean, we can use the formula:
Confidence interval = sample mean ± (critical value * standard deviation / square root of sample size)
First, we need to find the critical value from the t-distribution table. Since the sample size is 28, the degrees of freedom are 28 - 1 = 27. For a 95% confidence level and 27 degrees of freedom, the critical value is approximately 2.052.
Now we can calculate the confidence interval:
Confidence interval = 119 ± (2.052 * 14 / √28)
Standard deviation = 14
Sample size = 28
Calculating the square root of the sample size (√28):
√28 ≈ 5.29
Substituting the values into the formula:
Confidence interval = 119 ± (2.052 * 14 / 5.29)
Calculating the multiplication and division:
Confidence interval = 119 ± (28.728 / 5.29)
Calculating the division:
Confidence interval = 119 ± 5.43
Therefore, the 95% confidence interval for the population mean is approximately (113.57, 124.43).
To find a 95% confidence interval for the population mean, you can use the formula:
Confidence Interval = Sample Mean ± (Z * (Sample Standard Deviation / √Sample Size))
Where:
- Sample Mean is the mean of the sample (119 in this case).
- Sample Standard Deviation is the standard deviation of the sample (14 in this case).
- Sample Size is the number of observations in the sample (28 in this case).
- Z is the z-score corresponding to the desired confidence level (95% in this case).
To find the value of Z for a 95% confidence level, we can use a standard normal distribution table or a statistical calculator. The Z value for a 95% confidence level is approximately 1.96.
Now, let's substitute the given values into the formula:
Confidence Interval = 119 ± (1.96 * (14 / √28))
Calculating the above equation gives us:
Confidence Interval ≈ 119 ± (1.96 * (14 / √28))
Confidence Interval ≈ 119 ± (1.96 * (14 / 5.29))
Confidence Interval ≈ 119 ± (1.96 * 2.645)
Confidence Interval ≈ 119 ± 5.1812
Therefore, the 95% confidence interval for the population mean is approximately (113.82, 124.18).