A random sample of size 15 taken from a normally distributed population reveled a sample mean of 75 and a sample variance of 25. The upper limit of a 95% confidence interval for the population mean would equal?
77.77
To find the upper limit of a 95% confidence interval for the population mean, you can use the formula:
Upper Limit = Sample Mean + (Z * (Standard Deviation / √n))
Where:
- Sample Mean is the mean of the sample (given as 75 in this case).
- Z is the critical value corresponding to the desired level of confidence. For a 95% confidence level, the Z-value is approximately 1.96.
- Standard Deviation is the square root of the sample variance (given as √25 = 5 in this case).
- n is the sample size (given as 15).
Now let's calculate the upper limit:
Upper Limit = 75 + (1.96 * (5 / √15))
Upper Limit ≈ 75 + (1.96 * (5 / 3.873))
Upper Limit ≈ 75 + (1.96 * 1.29)
Upper Limit ≈ 75 + 2.52
Upper Limit ≈ 77.52
Therefore, the upper limit of the 95% confidence interval for the population mean is approximately 77.52.
To find the upper limit of a 95% confidence interval for the population mean, we can use the formula:
Upper limit = sample mean + (critical value * standard error)
First, let's calculate the critical value. Since we have a 95% confidence level, the significance level (alpha) is 1 - 0.95 = 0.05. Since the sample size is small (n = 15), we'll use a t-distribution instead of a standard normal distribution.
The degrees of freedom (df) for a small sample size is (n - 1) = 15 - 1 = 14. Since we want the two-tailed critical value, we divide the significance level by 2: alpha/2 = 0.05/2 = 0.025.
Next, we can consult a t-distribution table or use a calculator to find the critical value for a significance level of 0.025 and 14 degrees of freedom. Assuming a two-tailed test, the critical value is approximately 2.145.
Now, let's calculate the standard error. The standard error (SE) represents the standard deviation of the sampling distribution of the sample mean and is calculated by dividing the sample standard deviation (sample variance square root) by the square root of the sample size.
Standard error (SE) = sqrt(sample variance / sample size)
= sqrt(25/15)
= sqrt(5/3)
= 1.29 (rounded to two decimal places)
Now we can find the upper limit of the 95% confidence interval using the formula:
Upper limit = sample mean + (critical value * standard error)
= 75 + (2.145 * 1.29)
= 75 + 2.768
= 77.768 (rounded to three decimal places)
Therefore, the upper limit of the 95% confidence interval for the population mean is approximately 77.768.