Question

The true average diameter of ball bearings of a certain type is supposed to be 0.5 inch.

What conclusion is appropriate when testing H₀: μ = 0.5 versus H_a: μ ≠ 0.5 inch in each of the following situations?

(a)    

n = 15, t = 1.8, α = 0.05

1. Reject H₀

2.Fail to reject H₀    

(b)    

n = 15, t = −1.8, α = 0.05

1. Reject H₀

2. Fail to reject H₀    

(c)    

n = 27, t = −2.5,α = 0.01

1. Reject H₀

_2. Fail to reject H₀    

(d)    

n = 27, t = −3.7

1. Reject H₀ at any reasonable significance level

2. Fail to reject H₀ at any reasonable significance level    

Answer 1

Solution:
Given in the question
Null hypothesis H0: = 0.5
Alternative hypothesis Ha: 0.5
Solution(a)
n= 15, so df = n-1 = 15-1 = 14
t = 1.8
So from t table we found p-value at df =14 and this is two tailed test
p-value = 0.0934
So at alpha =0.05, we can see that p-value is greater than alpha value so we are failed to reject the null hypothesis.
Solution(b)
n= 15, so df = n-1 = 15-1 = 14
t = -1.8
So from t table we found p-value at df =14 and this is two tailed test
p-value = 0.0934
So at alpha =0.05, we can see that p-value is greater than alpha value so we are failed to reject the null hypothesis.
Solution(c)
n = 27, so df = 27-1 = 26
t = -2.5
So from t table we found p-value at df =26 and this is two tailed test
p-value = 0.0191
So at alpha =0.01, we can see that p-value is greater than alpha value so we are failed to reject the null hypothesis.
Solution(d)
n = 27, df = 27- 1 = 26
t = -3.7
So from t table we found p-value at df = 26 and this is two tailed test
P-value = 0.001.
So at alpha =0.01, we can see that the p-value is less than alpha value so we can reject the null hypothesis.
So reject H0 at any reasonable significance level.