Answer:
(a) Null hypothesis: The breaking strength of a fiber is required to be 150 psi
Alternate hypothesis: The breaking strength of a fiber is required to be 150 psi or greater than 150 psi
(b) Fail to reject the null hypothesis. The breaking strength of a fiber is required to be 150 psi
(c) P-value is 0.025
(d) 95% confidence interval for the mean breaking strength is between 145.227 psi and 154.773 psi
Step-by-step explanation:
(a) The null hypothesis is a statement from the population parameter which is subject to testing. The alternate hypothesis is also a statement from the population parameter that negates the null hypothesis.
(b) Z = (sample mean - population mean) ÷ sd/√n
sample mean = 145+153+150+147/4 = 148.75 psi, population mean = 150 psi, sd = 3 psi, n= 4
Z = (148.75 - 150) ÷ 3/√4 = -1.25 ÷ 1.5 = -0.833
This test is a two tailed test. The critical value at 0.05 significance level is 1.96. The region of no rejection of the null hypothesis lies between -1.96 and 1.96 since it is a two tailed test.
Conclusion: since -0.833 falls within the region bounded by -1.96 and 1.96, fail to reject the null hypothesis. The breaking strength of a fiber is required to be 150 psi
(c) The cumulative area for the critical value Z = 1.96 is 0.9750
P-value = 1 - 0.9750 = 0.025
(d) Confidence Interval = mean + or - error margin (E)
E = t×sd/√n
n = 4, degree of freedom = n-1 = 4-1 = 3, t-value corresponding to 3 degrees of freedom and 95% confidence level is 3.182
E = 3.182×3/√4 = 4.773
Lower limit = mean - E = 150 - 4.773 = 145.227
Upper limit = mean + E = 150 + 4.773 = 154.773
95% confidence interval is (145.227 psi, 154.773 psi)
