Question

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 41, with sample mean x = 45.4 and sample standard deviation s = 6.0.

(a) Compute a 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

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(b) Compute a 99% confidence interval for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

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(d) Now consider a sample size of 71. Compute a 99% confidence interval for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

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(e) Compute a 99% confidence interval for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

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(f) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

Answer 1

Refer t-table or use excel function "=T.INV.2T(0.01,40)" to find the value of tc

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Refer Standard normal table/Z-table or use excel function "=NORM.S.INV((1-0.005))" to find the Z-value

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Refer t-table or use excel function "=T.INV.2T(0.01,70)" to find the value of tc

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Refer Standard normal table/Z-table or use excel function "=NORM.S.INV((1-0.005))" to find the Z-value

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Yes, The respective interval based on the t distribution are longer.

As the sample size increases, the difference between the two methods becomes greater.