Question

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ. (Notice that, When σ is unknown and the sample is of size n < 30, there is only one method for constructing a confidence interval for the mean by using the Student's t distribution with d.f. = n - 1.) 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 = 30, with sample mean x = 45.2 and sample standard deviation s = 5.3.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% Lower limit and Upper limit

95% Lower limit and Upper limit

99% Lower limit and Upper limit

(d) Now consider a sample size of 50. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

90% Lower limit and Upper limit

95% Lower limit and Upper limit

99% Lower limit Upper limit

Please show me how to do this on a TI 84 Calculator if possible. Thank you!

Answer 1

Solution:-

a) 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution.

90% confidence intervals for μ is C.I = (43.556, 46.844).

C.I = 45.2 + 1.699*0.96764

C.I = 45.2 + 1.64403

C.I = (43.556, 46.844)

95% confidence intervals for μ is C.I = ( 43.220, 47.179).

C.I = 45.2 + 2.046*0.96764

C.I = 45.2 + 1.97979

C.I = ( 43.220, 47.179)

99% confidence intervals for μ is C.I = ( 42.532, 47.868).

C.I = 45.2 + 2.757*0.96764

C.I = 45.2 + 2.66778

C.I = ( 42.532, 47.868)

d) 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution.

90% confidence intervals for μ is C.I = (43.943, 46.457).

C.I = 45.2 + 1.677*0.74953

C.I = 45.2 + 1.25697

C.I = (43.943, 46.457)

95% confidence intervals for μ is C.I = (43.693, 46.707).

C.I = 45.2 + 2.01*0.74953

C.I = 45.2 + 1.5066

C.I = (43.693, 46.707)

99% confidence intervals for μ is C.I = ( 43.191, 47.209).

C.I = 45.2 + 2.68*0.74953

C.I = 45.2 + 2.009

C.I = ( 43.191, 47.209)