Question

1. (a) A statistician randomly sampled 100 observations and found

= 106 and s = 35. Calculate the t-statistic and p-value for testing

H₀: μ = 100 vs H_A: μ > 100.

Carry out the test at the 1% level of significance.

(b) Repeat part (a), with s = 25.

(c) Repeat part (a), with s = 15.

(d) Discuss what happens to the t-statistic and the p-value when the standard deviation decreases.

2. Repeat Question 1 using H_A: μ ≠ 100.

Answer 1

1a) H₀: μ = 100 vs H_A: μ > 100.

alpha=0.01

t= xbar-mean/s/sqrt(n)

t= 106-100/35/sqrt(100)

t= 6/35/10

t= 6/3.5

t= 1.71

d.f= 100-1=99

The p-value is .045199.

Since P value is GREATER than the level of significance hence NOT SIGNIFICANT therefore DO NOT REJECT NULL HYPOTHESIS H0.

b) s= 25

t= 106-100/25/sqrt(100)

t= 6/25/10

t= 6/2.5

t= 2.4

The p-value is .009132.

The result is significant at p < .01

Since P value is SMALLER than the level of significance hence SIGNIFICANT therefore REJECT NULL HYPOTHESIS H0.

c) s=15

t= 106-100/15/sqrt(100)

t= 6/15/10

t= 6/1.5

t=4

The p-value is .000061.

The result is significant because p < .01.

Since P value is SMALLER than the level of significance hence SIGNIFICANT therefore REJECT NULL HYPOTHESIS H0.

d) the t-statistic increasing while the p-value is decreasing when the standard deviation decreases.

NOTE: As per guidelines I have done the first part please re post the second question. Thank you.