Answer to Question #346791 in Statistics and Probability for Joyce

Question

It is claimed that the average weight of a bag of biscuits is 250 grams with a standard deviation of 20.5 grams. Would you agree to this claim if a random sample of 30 bags of biscuits showed an average weight of 240 grams, What is the tabular value if the level of significance is 5%?

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:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{240-250}{20.5/\sqrt{30}}=-2.6818$$

Since it is observed that $|z|=2.6818>1.96=z_c,$ it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is $p=2P(z<-2.6818)=0.007323,$ and since $p= 0.007323<0.05=\alpha,$ 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.