Question

The table below gives two samples selected from 10 supermarkets of the weekly sales of a popular soft drink. The first sample gives the details for a normal shelf display of the product, while the second sample gives the details for an end-aisle shelf display. Assuming equal variances, establish, at the 5% level of significance, whether there is a statistically significant difference in the mean weekly sales for the two display locations.
Normal display :22,34,52,62,30,40,64,84,56,59
END AISLE DISPLAY: 52,71,76,54,67,83,66,90,77,84

Answer 1

Answer on Question #48984 – Math – Statistics and Probability

The table below gives two samples selected from 10 supermarkets of the weekly sales of a popular soft drink. The first sample gives the details for a normal shelf display of the product, while the second sample gives the details for an end-aisle shelf display. Assuming equal variances, establish, at the 5% level of significance, whether there is a statistically significant difference in the mean weekly sales for the two display locations.

Normal display: 22,34,52,62,30,40,64,84,56,59

END AISLE DISPLAY: 52,71,76,54,67,83,66,90,77,84

Solution

$$H_0: \mu_1 = \mu_2; H_1: \mu_1 \neq \mu_2.$$$$\bar{x} = \frac{\sum x_i}{n} = \frac{503}{10} = 50.3.$$ $$\bar{y} = \frac{\sum y_i}{n} = \frac{720}{10} = 72.$$ $$s_{\bar{x}}^2 = \frac{\sum x_i^2 - n\bar{x}^2}{n-1} = \frac{28457 - 10 \cdot 50.3^2}{9} = 350.7.$$ $$s_{\bar{y}}^2 = \frac{\sum y_i^2 - n\bar{y}^2}{n-1} = \frac{53256 - 10 \cdot 72^2}{9} = 157.3.$$

Assuming equal variances, the t Statistic:

t = \frac{\bar{y} - \bar{x}}{\sqrt{\frac{(n-1)s_{\bar{x}}^2 + (n-1)s_{\bar{y}}^2}{(n-1) + (n-1)}} = \frac{\bar{y} - \bar{x}}{\sqrt{\frac{s_{\bar{x}}^2 + s_{\bar{y}}^2}{2}}} = \frac{72 - 50.3}{\sqrt{\frac{350.7 + 157.3}{2}}} = 1.36.

Critical value for $n - 1 = 10 - 1 = 9$ degrees of freedom and 5% level of significance is $t^* = 2.262$.

Critical region: $t > 2.262$.

We don't reject $H_0$ because test statistic $t = 1.36 < 2.262$. There is no statistically significant difference in the mean weekly sales for the two display locations.

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