Question

Suppose the mean Verbal SAT score for the population of first graders in New York City is 520, with a standard deviation of 95. An investigator believes that the mean Verbal SAT scores of first year Buying and Merchandising majors is significantly different from the mean of the population. The mean of a sample of 36 first year Buying and Merchandising majors is 548.Use the six-step approach to test the investigator’s prediction at the .05 level of significance.

NOTE: The steps must be clearly and neatly indicated. (2, 5, 3, 8, 3, 4 = 25 points)

Answer 1

Solution:

Step 1:

The null and alternative hypotheses are as follows:

H_{​​​​​​0} : μ = 520 i.e. The population mean of the Verbal SAT scores of first year Buying and Merchandising majors is 520.

H_{​​​​​1} : μ ≠ 520 i.e. The population mean of the Verbal SAT scores of first year Buying and Merchandising majors is not equal to 520.

Step 2:

To test hypothesis we shall use one sample z-test for mean. The test statistic is given as follows:

Where, x̄ is sample mean, μ is hypothesized value of population mean, σ is population standard deviation and n is sample size.

We have, x̄ = 548, μ = 520, σ = 95 and n = 36

The value of the test statistic is 1.76842.

Step 3:

Since, our test is two-tailed test, therefore we shall obtain two-tailed p-value for the test statistic. The two-tailed p-value is given as follows:

p-value = 2.P(Z > value of the test statistic)

Hence, p-value = 2.P(Z > 1.76842)

p-value = 0.07699

The p-value is 0.07699.

Step 4:

The specified significance level is 0.05

Step 5:

We make decision rule as follows:

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H_{​​​​​​0}) at given significance level.

If p-value is less than the significance level, then we reject the null hypothesis (H_{​​​​​​0}) at given significance level.

We have, p-value = 0.07699 and significance level = 0.05

(0.07699 > 0.05)

Since, p-value is greater than the significance level of 0.05, therefore we shall be fail to reject the null hypothesis (H_{​​​​​​0}) at 0.05 significance level.

Step 6 :

Conclusion : At significance level of 0.05, there is not sufficient evidence to support the investigator claim that the mean Verbal SAT scores of first year Buying and Merchandising majors is significantly different from the mean of the population.

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