Question

Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal.

Refer to the accompanying data set. Use a 0.05 significance level to test the claim that the sample of home voltages and the sample of generator voltages are from populations with the same mean. If there is a statistically significant​ difference, does that difference have practical​ significance?

Day	Home( volts)	Generator( volts)		Day	Home (volts)	Generator (volts)
1	123.6	124.4		21	123.8	124.7
2	123.7	124.5		22	123.9	124.8
3	124.0	124.6		23	123.6	124.1
4	123.1	124.4		24	123.7	124.6
5	123.8	124.6		25	123.1	124.6
6	123.4	124.2		26	123.2	124.2
7	123.4	125.1		27	124.0	124.8
8	123.2	125.0		28	123.9	124.1
9	123.4	124.1		29	123.1	124.9
10	124.0	124.7		30	123.0	124.7
11	123.2	124.8		31	123.4	124.7
12	124.0	124.9		32	123.3	124.3
13	123.9	124.4		33	123.1	124.0
14	123.7	124.7		34	123.6	124.2
15	123.2	124.2		35	123.5	124.8
16	124.0	124.5		36	123.3	124.7
17	124.1	124.3		37	123.8	124.9
18	123.2	124.3		38	123.8	124.5
19	123.7	125.0		39	123.1	125.0
20	123.4	124.8		40	123.8	124.8

Calculate the test statistic:

Find the P-value (round to 4 decimal places):

Make a conclusion about the null hypothesis and a final conclusion that addresses the original claim.

(Reject/Fail to reject) H0. There (Is/Is not) sufficient evidence to warrant rejection of the claim that the sample of home voltages and the sample of generator voltages are from populations with the same mean. The difference (Is/Is not) statistically significant.

If there is a statistically significant difference, does that difference have practical significance?

a.) The sample means suggest that the difference does not have practical significance. The generator could be used as a substitute when needed.

b.) The sample means suggest that the difference does not have practical significance. The generator could not be used as a substitute when needed.

c.) The sample means suggest that the difference does have practical significance. The generator could not be used as a substitute when needed.

d.) The difference is not statistically significant.

Answer 1

Let µ1 and µ2 be the population means of the home voltages and the generator voltages.

Here are the population SDs are unknown and also its assumed to be different.

To test H0:  µ1= µ2 vs H1:µ1≠µ2

The test statistic under H0 is

(X1bar-X2bar)/√(s1^2/n1+s^2/n2) ~t distribution with (n1+n2-2) degrees of freedom.

where X1bar=sample mean of home voltages.

X2bar =sample mean of generator voltages.

s1^2=sample variance of home voltages.

s2^2=sample variance of generator voltages.

n1=sample size of home voltages=40

n2=sample size of generator voltages=40

From the given data and using calculator,

X1bar=∑x1i/n=4942/40=123.55

s1^2=∑(x1i-x1bar)^2/(n-1)=0.1092 (Found from calculator)

X2bar=∑x2i/n=4982.9/40=124.5725

s2^2=∑(x2i-x2bar)^2/(n-1)=0.0882(Found from calculator)

So the test statistic =(X1bar-X2bar)/√(s1^2/n1+s^2/n2)

=(123.55-124.5725)/√(0.1092/40+0.0882/40)=-1.0225/0.0702=-14.5655

For,the two-tailed test and for t=-14.5655 and degrees of freedom =n1+n2-2=78, the p-value=0.0000 which is very small and is less than alpha=0.05.

Decision:Hence, we reject the null hypothesis at level of significance 0.05.

Conclusion:So we conclude that there are significant differences between the means of home voltages and the generator voltages.

There is sufficient evidence to warrant rejection of the claim that the sample of home voltages and the sample of generator voltages are from populations with the same mean.

The difference IS statistically significant.

So the answer is: c.) The sample means suggest that the difference does have practical significance. The generator could not be used as a substitute when needed.