Question

			Sigma known
Ratings			Population Standard Deviation
			2.64
Mean	42.95384615		95% CI Margin of Error
Standard Error	0.327747828		0.641805331
Median	43		95% Confidence Interval
Mode	44		42.31204082	43.59565148
Standard Deviation	2.642387469		99% CI Margin of Error
Sample Variance	6.982211538		0.843188126
Kurtosis	-0.392154755		99% Confidence Interval
Skewness	-0.446608112		42.11065803
Range	12		43.79703428
Minimum	36
Maximum	48
Sum	2792		Sigma Unknown
Count	65		95% Confidence Interval
Confidence Level(95.0%)	0.654751556		42.2990946	43.60859771

1. State the 95% confidence interval for sigma unknown.
2. What is the multiplier used by Excel if sigma is unknown?
3. What do you notice about the 95% Confidence Intervals when sigma is known versus unknown? Explain.

Answer 1

State the 95% confidence interval for sigma unknown.

From the output

ans: the 95% confidence interval for sigma unknown is (42.2991, 43.6086) (rounded to 4 decimals)

What is the multiplier used by Excel if sigma is unknown?

When sigma (the population standard deviation) is unknown, the sampling distribution of mean is assumed to have t distribution (We do not apply the central limit theorem and we assume that the population is normally distributed)

The significance level for 95% confidence level is

The right tail critical value is

The sample size is n=65 (Given by the count). The degrees of freedom are n-1=65-1=64.

Using the Excel formula =T.INV.2T(0.05,64), we get the value

95% confidence interval is

where

is the sample mean

is the multiplier

is the standard error of mean

Ans: the multiplier used by Excel if sigma is unknown is 1.9977 (rounded to 4 decimals)

What do you notice about the 95% Confidence Intervals when sigma is known versus unknown? Explain.

using this

The 95% confidence interval when sigma is know is (42.3120, 43.5957)

The width/length of the 95% confidence interval when sigma is known is 43.5957-42.3120=1.2836
The width/length of the 95% confidence interval when sigma is unknown is 43.6086-42.2991=1.3095

ans: We can see that the 95% confidence interval when sigma is unknown is wider than when sigma is known. This is due to the fact that when sigma is unknown, we use a larger multiplier (1.9977), compared to when sigma is known (multiplier is 1.96). The wider interval when sigma is unknown also indicates that we are less accurate about the interval in which the population mean lies, when we do not know the population standard deviation (sigma is unknown) in comparison to when we do know the population standard deviation (sigma is known)