Question

The annual incomes for a random sample of 16 engineers working in Dubai have a standard deviation of $14,900. The annual incomes for a random sample of 17 engineers working in Abu Dhabi have a standard deviation of $9600. Using this information, can you conclude that the standard deviation of the annual incomes for engineers is greater in Dubai than in Abu Dhabi? Use = 0:05:

a. Identify the claim and state H0 and Ha: b. Find the critical value and identify the rejection region. c. Find the standardized test statistic. d. Decide whether to reject the null hypothesis. e. Interpret the decision in the context of the original claim.

Answer 1

Answers :

a)

Null Hypothesis Ho : Variance of the annual incomes for engineers in Dubai and Abu Dhabi are same.

Alternate Hypothesis Ha : Variance of the annual incomes for engineers in Dubai is greater than in  Abu Dhabi.

b)

An F-test  is used to test if the variances of two populations are equal.

Formula of F test as given below;

F = Variance of Dubai engineers / Variance of Abu Dhabi engineers = Var1/Var2 ................(1)

To find critical value :

Given values are as below;

n1 = 16 ( Sample size of Dubai engineers )

n2 = 17 ( Sample size of Abu Dhabi engineers )

Var1 = 14900 square ( Variance of Dubai engineers )

Var2 = 9600 square  ( Variance of Abu Dhabi engineers )

Alpha = 0.05

F critical value = 3.24 ( from below F table )

Probability p Table entry for p is the critical value F with probability p lying to its right. F TABLE E F critical values Degrees of freedom in the numerator 10 12 15 20 25 30 40 50 60 120 1000 100 .050 UZS ,010 .001 3.05 4.49 6.12 8.53 16 2.67 3.63 4.69 6.23 10.97 2.46 3.24 4.08 5.29 9.01 2.33 3.01 3.73 4.77 7.94 2.24 2.85 3.50 4.44 7.27 2.18 2.74 3.34 4.20 6.80 2.13 2.66 3.22 4.03 6.46 2.09 2.59 3.12 3.89 6.19 2.06 2.54 3.05 3.78 5.98 16.12

Identifying the Rejection region as below ;

F distribution curve When F statistic value goes beyond F critical value → Reject Ho Do not reject Ho Reject Ho 0 F-critical value =3.24

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c.

F statistic value = Var1 / Var2 = 14900 Square / 9600 square = 2.4090  

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d.

Decision :

Since F statistic value = 2.4090 < F critical value = 3.24, we ACCEPT Ho.

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e.

Interpretation:

Since we ACCEPT Ho, the conclusion is as below ;

Variance of the annual incomes for engineers in Dubai and Abu Dhabi are same.

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