Question

Using the data below, test the null hypothesis that the variance in the Close Ratio for the sample of employees is at most 1.5% at 5% significance level.

Years Employed	Gender	Certified	Prior Background	Close Ratio	Sales
7	M	Y	Y	44.40%	$   201,500
8	M	Y	Y	54.20%	$   129,000
4	M	Y	Y	60.70%	$   101,100
8	M	N	N	42.60%	$      99,700
6	M	Y	N	68.30%	$      98,100
6	M	N	N	37.90%	$      93,900
2	M	N	N	29.50%	$      90,800
2	M	N	N	42.40%	$      90,600
7	M	Y	Y	37.70%	$      89,200
2	M	N	N	35.00%	$      86,600
2	M	N	N	45.80%	$      83,900
3	M	N	N	50.70%	$      80,000
4	M	Y	Y	29.80%	$      77,300
7	M	N	N	38.70%	$      67,100
5	M	Y	Y	62.20%	$      58,900
3	M	Y	Y	63.20%	$      56,600
6	M	N	N	18.60%	$      54,700
3	F	Y	Y	32.00%	$   126,300
4	F	Y	Y	50.00%	$      95,800
9	F	Y	Y	51.70%	$      93,100
2	F	N	Y	29.20%	$      91,600
2	F	N	N	58.20%	$      89,000
4	F	Y	Y	44.00%	$      82,300
8	F	Y	Y	67.70%	$      81,000
6	F	N	Y	61.20%	$      78,200
7	F	N	Y	54.40%	$      71,300
8	F	N	Y	38.60%	$      67,500
2	F	Y	Y	40.40%	$      64,800
4	F	N	Y	41.50%	$      57,200
2	F	N	Y	40.70%	$      54,100
2	F	N	Y	36.30%	$      51,700

Answer 1

Variance=1.5 implies SD=1.224744871

Here we are to test

Here the sample size n=31

Let the sample mean be and the parent population from which the sample is drawn follows Normal distribution.

The test statistic is given by

and under

Now for the calculation of the statistic, the following table is made:

Close Ratio ( in %)
44.4	3.524703
54.2	62.76728
60.7	208.0108
42.6	13.52341
68.3	484.9941
37.9	70.18115
29.5	281.4818
42.4	15.03438
37.7	73.57212
35	127.1802
45.8	0.227929
50.7	19.55922
29.8	271.5053
38.7	57.41728
62.2	253.5286
63.2	286.3737
18.6	766.0395
32	203.8447
50	13.85761
51.7	29.40438
29.2	291.6383
58.2	142.1479
44	5.186639
67.7	458.927
61.2	222.6834
54.4	65.97632
38.6	58.94277
40.4	34.54406
41.5	22.82374
40.7	31.10761
63.3	289.7683
Total	4865.774

Hence the test statistic is obtained as

The critical value, as obtained from the Biometrika table, is  

As the observed value is greater than that the critical value, we reject the null hypothesis at 5% level of significance and hence conclude that the variance of the close ratio is more than 1.5% at 5% level of significance.

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