Answer to Question #143269 in Statistics and Probability for Auguste Nkole

Question

QUESTION 1 (14 MARKS)

A specific task in a factory takes 5 minutes in average to be completed. The factory manager believes that the workers spend more time in this operation. The manager selects a sample of 11 timings for this worker and collects the following data (in minutes): 4.8 ; 5.6 ; 5.3 ; 5.2 ; 4.9 ; 4.7 ; 5.7 ; 4.9 ; 5.7 ; 4.9 ; 4.6.

To test the validity of this factory manager’s belief, and using α = 0.05, Answer the following questions

1.1) Formulate the null and alternative hypotheses (2)

1.2) Select and compute the test statistic (6)

1.3) Obtain the critical value and formulate the decision. (3)

1.4) Based on the information obtained in Questions 1.1 ; 1.2 & 1.3, what can you say about the manager’s belief ? (3)

Answer 1

$n = 11$

$\alpha = 0.05$

1) A specific task in a factory takes 5 minutes in average to be completed and the factory manager believes that the workers spend more time in this operation, hence:

$H_0: \mu=5$

$H_1: \mu>5$ (one-tailed)

2) A sample size is less than 30 and we do not know the population standard deviation, therefore this is a t-test. Test statistic:

$$t_c=\frac{\bar x - \mu_0}{\frac{\sigma}{\sqrt{n}}}$$

$\mu_0=5$

$\bar x = \frac{\sum x}{n}=\frac{4.8+5.6+5.3+5.2+4.9+4.7+5.7+4.9+5.7+4.9+4.6}{11}\approx5.1181$

$\sigma=\sqrt{\frac{\sum{(x-\bar x)^2}}{n-1}}\approx0.4045$

$t_c=\frac{5.1181 - 5}{\frac{0.4045}{\sqrt{11}}}=0.968$

3) Degrees of freedom are n – 1 or 10. Looking up 10 degrees of freedom and 0.05 level of significance of the t-table we find 1.812. This is the critical value.

Critical region is t>1.812.

$t_c$ does not lie in the critical region and thus we accept the null hypothesis.

4) There is no evidence at the 0.05 level of significance to support the factory manager's belief.