Answer to Question #303793 in Statistics and Probability for bridgette

Question

1.It is claimed that the average weight of babies at birth is 3.4 kg. The average weight of a random sample of 30 newly born babies was determined.It was found out that the average weight was 3.1 kg. Is there a reason to believe that the average weight of babies at birth is not 3.4 kg? Assume that the population standard deviation is 1.1kg. Use 0.05 level of significance.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=3.4$

$H_1:\mu\not=3.4$

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=29$ degrees of freedom, and the critical value for a two-tailed test is $t_c = 2.04523.$

The rejection region for this two-tailed test is $R = \{t: |t| > 2.04523\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}=\dfrac{3.1-3.4}{1.1/\sqrt{30}}=-1.49379$$

Since it is observed that $|t| = 1.49379\le 2.04523=t_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for two-tailed, $df=29$ degrees of freedom, $t=-1.49379,$ is $p=0.146032,$ and since $p=0.146032>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$ is different than $3.4,$ at the $\alpha = 0.05$ significance level.