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.

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.