Question

The breaking strengths of cables produced by a certain manufacturer have historically had a mean of 1875 pounds and a standard deviation of 75 pounds. The company believes that, due to an improvement in the manufacturing process, the mean breaking strength of the cables is now greater than 1875 pounds. To see if this is the case, 29 newly manufactured cables are randomly chosen and tested, and their mean breaking strength is found to be 1892 pounds. Assume that the population is normally distributed. Can we support, at the 0.01 level of significance, the claim that the population mean breaking strength of the newly-manufactured cables is greater than 1875 pounds? Assume that the population standard deviation has not changed.

Perform a one-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places, and round your responses as specified below

(a) State the null hypothesis and the alternative hypothesis

(b) Determine the type of test statistic to use

(c) Find the value of the test statistic

(d) Find the p-value

(e) Can we support the claim that the population mean breaking strength of the newly-manufactured cables is greater than 1875 pounds?

Answer 1

In this hypothesis test, the null hypothesis states that the population mean breaking strength is equal to 1875 pounds, while the alternative hypothesis states that it is greater. Using a one-tailed z-test, the calculated test statistic is approximately 1.576. The corresponding p-value is approximately 0.0587, which is greater than the significance level of 0.01. Therefore, we do not have enough evidence to support the claim that the population mean breaking strength of the newly-manufactured cables is greater than 1875 pounds.

Step-by-step explanation:

(a) The null hypothesis (H0) is that the population mean breaking strength of the newly-manufactured cables is equal to 1875 pounds. The alternative hypothesis (H1) is that the population mean breaking strength of the newly-manufactured cables is greater than 1875 pounds.

(b) Since we are performing a one-tailed test and have the population standard deviation, we can use the z-test statistic.

(c) The test statistic (z-score) can be calculated using the formula:

z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

Substituting the given values:

z = (1892 - 1875) / (75 / sqrt(29))

Calculating this expression gives us a z-score of approximately 1.576.

(d) To find the p-value, we need to find the area under the standard normal distribution curve to the right of the calculated z-score. Using a standard normal distribution table or a calculator, we find that the p-value is approximately 0.0587.

(e) Since the p-value (0.0587) is greater than the significance level (0.01), we do not have enough evidence to support the claim that the population mean breaking strength of the newly-manufactured cables is greater than 1875 pounds.