Question

The manufacturer of an MP3 player wanted to know whether a 10% reduction in price is enough to increase the sales of its product. To investigate, the owner randomly selected eight outlets and sold the MP3 player at the reduced price. At seven randomly selected outlets, the MP3 player was sold at the regular price. Reported below is the number of units sold last month at the regular and reduced prices at the randomly selected outlets. Regular price 138 124 89 112 116 123 98 Reduced price 124 134 154 135 118 126 133 132 i. Compute the pooled estimate of the variance. (Round your answer to 3 decimal places.)
ii. Compute the test statistic. (Round your answer to 2 decimal places.)
iii. State your decision about the null hypothesis.

Answer 1

The pooled estimate of the variance is 187.650 and the test statistic is -2.32

Pooled estimate of the variance

The dataset is given as:

Regular price 138 124 89 112 116 123 98

Reduced price 124 134 154 135 118 126 133 132

Let the regular price be dataset 1 and the reduced price be dataset 2.

So, we have:

#1: 138 124 89 112 116 123 98

#2: 124 134 154 135 118 126 133 132

Calculate the sample means and the sample standard deviations using a graphing calculator.

#1

$\sigma_1 = 16.56$

$\bar x_1 = 114.29$

$\sigma_1^2 = 274.24$

#2

$\sigma_2 = 10.65$

$\bar x_2 = 132$

$\sigma_2^2 = 113.43$

The pooled estimate of the variance is:

$\sigma_p^2 = \frac{\sigma_1^2(n_1 - 1) + \sigma_2^2(n_2 - 1)}{(n_1 - 1) + (n_2 - 1)}$

This gives

$\sigma_p^2 = \frac{274.24 * (7 - 1) + 113.43 * (8 - 1)}{(7- 1) + (8- 1)}$

Evaluate the factors

$\sigma_p^2 = \frac{2439.45}{13}$

Divide

$\sigma_p^2 = 187.65$

Hence, the pooled estimate of the variance is 187.650

The test statistic

This is calculated using:

$t = \frac{\bar x_1 - \bar x_2}{\sqrt{\sigma_p^2} * \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

This gives

$t = \frac{114.29 - 132}{\sqrt{187.650} * \sqrt{\frac{1}{7} + \frac{1}{6}}}$

This gives

$t = \frac{-17.71}{\sqrt{187.650} * \sqrt{\frac{13}{42}}}$

Evaluate the product

$t = \frac{-17.71}{\sqrt{58.08214}}$

Evaluate the exponent

$t = \frac{-17.71}{7.62}$

Divide

t = -2.32

Hence, the test statistic is -2.32

The decision about the null hypothesis

The critical value at 0.025 significance level is -1.96

-2.32 is less than -1.96

This means that we accept the null hypothesis

Read more about test statistic at:

brainly.com/question/14128303

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