Question

The breaking strengths of cables produced by a certain manufacturer have a mean, μ , of 1900 pounds, and a standard deviation of 100 pounds. It is claimed that an improvement in the manufacturing process has increased the mean breaking strength. To evaluate this claim, 21 newly manufactured cables are randomly chosen and tested, and their mean breaking strength is found to be 1959 pounds. Assume that the population is normally distributed. Can we support, at the 0.1 level of significance, the claim that the mean breaking strength has increased? (Assume that the standard deviation has not changed.) Perform a one-tailed test. Then fill in the table below. Carry your intermediate computations to at least three decimal places, and round your responses as specified in the table. (If necessary, consult a list of formulas.) The null hypothesis: H 0 : The alternative hypothesis: H 1 : The type of test statistic: The value of the test statistic: (Round to at least three decimal places.) The p-value: (Round to at least three decimal places.) Can we support the claim that the mean breaking strength has increased? Yes No μ σ p x s p

Answer 1

Solution:

Here, we have to use one sample z test for the population mean. The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The mean breaking strength has not increased.

Alternative hypothesis: H_a: The mean breaking strength has increased.

Null hypothesis: H₀: µ = 1900

Alternative hypothesis: H_a: µ > 1900

This is an upper tailed or right tailed (one tailed) test.

We are given

Level of significance = α = 0.10

Population standard deviation = σ = 100

Sample size = n = 21

Sample mean = Xbar = 1959

The test statistic formula for this test is given as below:

Z = (Xbar - µ) / [σ/sqrt(21)]

Z = (1959 – 1900) / [100/sqrt(21)]

Z = 59/ 21.8218

Z = 2.7037

Test statistic = 2.7037

P-value = 0.0034

(by using z-table)

Critical value = 1.2816

(by using z-table)

P-value < α = 0.10

So, we reject the null hypothesis

There is sufficient evidence to conclude that the mean breaking strength has increased.

Can we support the claim that the mean breaking strength has increased?

Answer: Yes, there is sufficient evidence to conclude that the mean breaking strength has increased.