Question

Taylor’s company does filling operation in a ground-beef packing plant. Three filling operations are studied by a research team. The first random sampling contained 25 packages resulting means and standard deviations were 1405.5 g and 11.2 g. The second random sampling contained 45 packages resulting means and standard deviations were 1410 g and 10.5 g and lastly the third random sampling contained 50 packages resulting means and standard deviations were 1412.5 g and 9.5 g.

a. Using a .05 significance level, is there sufficient evidence to indicate that the true mean weight of the packages differs for the first and second rounds of packaging?

b. Using a .05 significance level, is there sufficient evidence to indicate that the true mean weight of the packages the second round is smaller than the third rounds of packaging?

Answer 1

Given,

$\small \mu_1 =1405.5$,   $\small s_1 =11.2$,  

1 = 25

112= 1410
,
82 = 10.5
,   $\small n_2 = 45$

$\small \mu_3 =1412.5$,   $\small s_3 =9.5$,   $\small n_3 = 50$

a.

The hypothesis is,

$\small H_0:\mu_1 = \mu_2$

$\small H_1:\mu_1 \neq \mu_2$

The test statistic is,

$\small t=\frac{x_{1} - x_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$

t = -1.6778

the corresponding p value is 0.0980

Since the p value is greater than the level of significance, we fail to reject the null hypotheisis. We conclude that the there is not enough evidence to suggest that the mean of first and second packaging is different.

b.

The hypothesis is,

$\small H_0:\mu_2 = \mu_3$

$\small H_1:\mu_2 < \mu_3$

The test statistic is,

$\small z=\frac{x_{1} - x_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}$

$\small z=-1.2120$

The corresponding p value is 0.1128

Since the p value is greater than the level of significance, we fail to reject the null hypotheisis. We conclude that the there is not enough evidence to suggest that the mean of second packaging is less than third packaging.