Question

. (a) The performance of staff is rated as excellent, satisfactory, pass or fail during the evaluation period. Suppose the staff can be categorized as technical staff and administrative staff. A random 3 sample of 1000 staff were selected with their performance evaluation tabulated below.

excellent satisfactory pass fail

technical staff 50 250 200 30

adminstrative staff 60 200 200 10

Test whether there is an association between the category of staff and their performance evaluation at 1% level of significance. [6marks]

(b) A random sample of 175 chocolate cookies from ”Chips Ahoy” were tested for the amount of chocolate chip content, measured as a percentage defined as the ratio of the weight of the chocolate chips over the total weight of the cookie. The information is collected below.

Chocolate Content (in %) Number of Cookies

26 ≤ x < 28 7

28 ≤ x < 30 22

30 ≤ x < 32 36

32 ≤ x < 34 45

34 ≤ x < 36 33

36 ≤ x < 38 28

38 ≤ x < 40 4

Is there any evidence that the percentage of chocolate chip content of ”Chips Ahoy” cookies is normally distributed, at 5% level of significance? [10 marks]

(c) ”I-Phonie” Company surveyed 1000 of their phone customers and asked them what is their favourite colours for their phone. The colours chosen by their customers surveyed are represented below:

black:100

white:150

red:200

blue: 50

green 100

yellow 80

pink 140

grey 180

At 1% level of significance, is there any evidence indicating that there are some colours which are more favourable? [4 marks]

Answer 1

(a) The hypothesis being tested is:

H0: There is no association between the category of staff and their performance evaluation

Ha: There is an association between the category of staff and their performance evaluation

		Col 1	Col 2	Col 3	Col 4	Total
Row 1	Observed	50	250	200	30	530
	Expected	58.30	238.50	212.00	21.20	530.00
	O - E	-8.30	11.50	-12.00	8.80	0.00
	(O - E)² / E	1.18	0.55	0.68	3.65	6.07
Row 2	Observed	60	200	200	10	470
	Expected	51.70	211.50	188.00	18.80	470.00
	O - E	8.30	-11.50	12.00	-8.80	0.00
	(O - E)² / E	1.33	0.63	0.77	4.12	6.84
Total	Observed	110	450	400	40	1000
	Expected	110.00	450.00	400.00	40.00	1000.00
	O - E	0.00	0.00	0.00	0.00	0.00
	(O - E)² / E	2.51	1.18	1.45	7.77	12.91
		12.91	chi-square
		3	df
		.0048	p-value

The p-value is 0.0048.

Since the p-value (0.0048) is less than the significance level (0.01), we can reject the null hypothesis.

Therefore, we can conclude that there is an association between the category of staff and their performance evaluation.

(b) The mean and standard deviation values are missing.

(c) The hypothesis being tested is:

H0: All colours are equally favourable

Ha: There are some colours which are more favourable

observed	expected	O - E	(O - E)² / E
100	125.000	-25.000	5.000
150	125.000	25.000	5.000
200	125.000	75.000	45.000
50	125.000	-75.000	45.000
100	125.000	-25.000	5.000
80	125.000	-45.000	16.200
140	125.000	15.000	1.800
180	125.000	55.000	24.200
1000	1000.000	0.000	147.200
147.20	chi-square
7	df
1.57E-28	p-value

The p-value is 0.0000.

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 there are some colours which are more favourable.