Question

2. Calculate the 99%, 95%, and 90% confidence interval for the following information. Identify how these confidence intervals are similar and how they are different. Explain why. (70 points)

x̄ = 55 s = 15 n = 101

1. The 99% Confidence Interval:

2. The 95% Confidence Interval:

3. The 90% Confidence Interval:

4. Similarities:

5. Differences:

6. Why?

Answer 1

Answer)

Given info

Mean = 55

S.d = 15

N = 101

1)

As the population s.d is not given and we are given with sample s.d as the best estimate we will use t distribution to estimate the interval

Degrees of freedom is = n-1 = 100

For 100 dof and 99% confidence level critical value t from t table is = 2.63

Margin of error (MOE) = t*s.d/√n = 2.63*15/√101 = 3.92

Interval is given by

(Mean - MOE, Mean + MOE)

[51.08, 58.92].

You can be 99% confident that the population mean (μ) falls between 51.08 and 58.92.

B)

Critical value t for 95% confidence level with 100 dof = 1.98

Moe = 1.98*15/√101 = 2.96

[52.04, 57.96].

You can be 95% confident that the population mean (μ) falls between 52.04 and 57.96.

C)

Critical value t for 90% confidence level with 100 dof = 1.66

Moe = t*s.d/√n = 1.66*15/√101 = 2.48

[52.52, 57.48].

You can be 90% confident that the population mean (μ) falls between 52.52 and 57.48.

Similarity is that they all give us the range in which population parameter may lie with certain confidence level

Differences is that

We can see that as the confidence level decreases, width of the interval also decreases.