Question

Write the null and alternative hypothesis of given problem: ( 10 points)

    1. It is claimed that the average weight of a bag of biscuit is 250 grams with a standard deviation of 20.5 grams. Would you agree to this claim if a random sample of 50 bags of biscuits showed an average weight of 240 grams, using a 5% level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=250$

$H_1:\mu\not=250$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ and the critical value for a two-tailed test is $z_c = 1.96.$

The rejection region for this two-tailed test is $R = \{z: |z| > 1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{240-250}{20.5/\sqrt{50}}\approx-3.4493$$

Since it is observed that $|z| = 3.449 > 1.96=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=2P(Z<-3.4493)\approx0.000562,$ and since $p=0.0006<0.05=\alpha,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is different than $250,$ at the $\alpha = 0.05$ significance level.