Answer:
(a) Null hypothesis: The mean breaking strength of a fiber is 150 psi.
Alternate hypothesis: The mean breaking strength of a fiber is less than or equal to 150 psi.
(b) The test statistic is -1.67. There is sufficient evidence to conclude that the mean breaking strength of a fiber is 150 psi.
(c) No, the P value should not be less than the significance level (alpha).
(d) 95% confidence interval for the true mean breaking strength is between a lower limit of 140.71 psi and an upper limit of 154.29 psi.
Step-by-step explanation:
(a) A null hypothesis is a statement from a population parameter which is either rejected or accepted (fail to reject) upon testing. It is expressed using the equality sign.
An alternate hypothesis is also a statement from the population parameter negates the null hypothesis and is accepted if the null hypothesis is rejected. It is expressed using any of the inequality signs.
(b) sample mean = (145+153+150+142)/4 = 590/4 = 147.5 psi
population mean = 150 psi
population sd (sigma) = 3 psia
n = 4
Test statistic (z) = (sample mean - population mean) ÷ (population sd/√n) = (147.5 - 150) ÷ (3/√4) = -2.5 ÷ 1.5 = -1.67
The test is a two-tailed test because the alternate hypothesis is expressed using less than or equal to. For a 5% significance level, critical values are -1.96 and 1.96.
Conclusion:
Fail to reject the null hypothesis because the test statistic -1.67 falls within the region bounded by the critical values.
(c) The P value should not be less than alpha because to fail to reject the null hypothesis using the P value approach, the P value must be greater than alpha (significance level).
(d) Confidence interval is given as mean +/- margin of error (E)
mean = 147.5 psi
sample sd = sqrt[((145-147.5)^2 + (153-147.5)^2 + (150-147.5)^2 + (142-147.5)^2) ÷ 4] = 4.27 psi
n = 4
degree of freedom = n - 1 = 4-1 = 3
confidence level = 95%
critical value (t) corresponding to 3 degrees of freedom and 95% confidence level is 3.182
E = t × sample sd/√n = 3.182×4.27/√4 = 6.79 psi
Lower limit = mean - E = 147.5 - 6.79 = 140.71 psi
Upper limit = mean + E = 147.5 + 6.79 = 154.29 psi
95% confidence interval is (140.71, 154.29)
