Statistical Techniques in Business and Economics
Statistical Techniques in Business and Economics
16th Edition
ISBN: 9780077639723
Author: Lind
Publisher: Mcgraw-Hill Course Content Delivery
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Chapter 11, Problem 12E
To determine

State whether it is reasonable to conclude that the mean weekly salary of nurses is higher.

Obtain the p-value.

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Answer to Problem 12E

No, there is no enough evidence to conclude that the mean weekly salary of nurses is higher.

The p-value is 0.244.

Explanation of Solution

It is expected that the mean weekly salary of nurses is higher.

Therefore, the test hypotheses are given below:

Denote μ1 as the mean weekly salary of nurses and μ2 as the mean weekly salary of school teachers.

Null hypothesis: H0:μ1μ2.

That is, the mean weekly salary of nurses are at most the mean weekly salary of school teachers.

Alternative hypothesis: Ha:μ1>μ2

That is, the mean weekly salary of nurses is greater than the mean weekly salary of school teachers.

In this context, the level of significance is 0.05.

Necessary assumptions required for using the formula:

  • The sampled populations are approximately normally distributed.
  • The two samples are independent.
  • The standard deviations for the two populations are equal.

In this context, the two populations are independent and distributed to normal. The population standard deviations are unknown.

Test statistic for two-sample test of means-unknownσ:

t=X¯1X¯2sP2(1n1+1n2)

Where, n1and n2 are the sample sizes of two population, s12ands22 are the sample variances, and sp2 is the pooled estimate of σ2.

Here,

 sp2=(n11)s12+(n21)s22n1+n22.

Excel procedure to find the mean and standard deviation for samples school teacher and nurses:

  • Enter the data values in the column H and column I.
  • Obtain the sample mean 1 (X¯1) in cell A1, enter the formula “=AVERAGE(H1:H8)”.
  • Press “Enter”.
  • Obtain the sample variance 1(s12)  in cell B1, enter the formula “=VAR.S(H1:H8)”.
  • Press “Enter”.
  • Obtain the sample mean 2 (X¯1) in cell A2, enter the formula “=AVERAGE(I1:I12)”.
  • Press “Enter”.
  • Obtain the sample variance 2 (s22) in cell B2, enter the formula “=VAR.S(I1:I12)”.
  • Press “Enter”.

Output obtained using EXCEL is given below:

Statistical Techniques in Business and Economics, Chapter 11, Problem 12E , additional homework tip  1

The pooled variance is obtained as given below:

Substitute s12 as 1,183.643, s22 as 521.66, n1 as 8, and n2 as 12.

sp2=(81)(1,183.643)+(121)(521.66)8+122=14,023.76118=779.0978

The test statistic is given below:

Substitute X¯1 as 835.75, X¯2 as 826.75, sp2 as 779.0978, n1 as 8, and n2 as 12.

t=X¯1X¯2sP2(1n1+1n2)=835.75826.75779.0978(18+112)=912.7402=0.7064

Rejection region:

In context, the level of the test, α, is 0.01.

Here, the alternative is the right-tailed test. Hence, the rejection region will be t>tα .

The critical value has to be obtained for t0.01,(n1+n22)=t0.01,18.

Critical value:

From the “Appendix B, Table B.5 Student’s t Distribution”, the critical value for 18 df for the level of significance 0.01 is 2.552.

Thus, the rejection region under the level of significance of 0.01 is t>2.552.

Decision rule:

  • If t>2.552, reject the null hypothesis.
  • Otherwise fail to reject the null hypothesis.

Conclusion:

Here, the test statistic, t, does not lie in the rejection region.

Therefore, by the decision rule, fail to reject the null hypothesis.

Therefore, there is no evidence to conclude that the mean weekly salary of nurses is higher.

p-value:

Step-by-step procedure to obtain the p-value using EXCEL:

  • In a cell A1, enter the formula “=T.DIST.RT(0.7064,18)”.
  • Press “Enter”.

Output obtained using EXCEL is given below:

Statistical Techniques in Business and Economics, Chapter 11, Problem 12E , additional homework tip  2

Thus, the p-value is 0.244.

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