Question

Because of high production change-over time and costs, a director of manufacturing must convince management that a proposed manufacturing method reduces costs before the new method can be implemented. The current production method operates with a mean cost of £320 per hour. A sample of 1,000 products have been analysed using the innovative manufacturing method. The director of manufacturing observed the mean cost was £300 per hour and a standard deviation of £20 per hour.

a) Define an appropriate system of hypotheses (null hypothesis and alternative hypothesis) to test whether the mean cost of production is reduced.

b) What assumption(s) about the population is/are necessary to perform this test?

c) What is the value of the test statistic?

d) What is the approximate p-value associated to the test statistic?

e) What is the critical value?

f) At the 1% significance level, what is your conclusion? If the level of significance was 5% would your conclusion be different? Explain your answer.

(g) Calculate the 95% approximate confidence interval for the population mean cost of production.

Answer 1

(a) The hypothesis being tested is:

H0: µ = 320

Ha: µ < 320

(b) The one-sample t-test has four main assumptions:

• • The dependent variable must be continuous (interval/ratio).
• • The observations are independent of one another.
• • The dependent variable should be approximately normally distributed.
• • The dependent variable should not contain any outliers.

(c) The test statistic, t = (x - µ)/s/√n

t = (x - µ)/s/√n = (300 - 320)/20/√1000 = -31.623

(d) p-value = 0.0000

(e) The critical value is 2.33.

(f) Since the p-value (0.0000) is less than the significance level (0.01), we can reject the null hypothesis.

Therefore, we can conclude that the mean cost of production is reduced.

Since the p-value (0.0000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the mean cost of production is reduced.

(g) 300 2.33*(20/√1000) = 298.76, 301.24

The 95% approximate confidence interval for the population mean cost of production is between 298.76 and 301.24.

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