Question

A random sample of 150 donations at a certain blood bank reveals that 70 were Type A blood. Does this suggest that the actual percentage of Type A donations differs from 40%, the percentage of the population having Type A blood? Carry out a test of hypotheses using a significance level of 0.01. Would your conclusion have been different if significance level of 0.05 had been used? Use only the P-Value Approach. State H0 and Ha. (20 pts)

Answer 1

`z=\frac{0.467 -0.4}{\sqrt{(0.4(1-0.4))/(150)}}=1.675`

Now we can calculate th p value using the alternative hypothesis with the following probability:

`p_v =2*P(z>1.675)=0.0939`

Since the p value is higher than the significance level given of 0.05 we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true proportion of Type A donations is not significantly different from 0.4 or 40% at 5% of significance.

Explanation:

Information given

n=150 represent the random sample taken

X=70 represent the samples with Type A blood

`\hat p=(70)/(150)=0.467` estimated proportion of samples with Type A blood

`p_o=0.4` is the value to verify

`\alpha=0.05` represent the significance level

zwould represent the statistic

`p_v` represent the p value

System of hypothesis

We want to verify if the actual percentage of Type A donations differs from 40%, then the system of hypothesis are:

Null hypothesis:
`p=0.4`

Alternative hypothesis:
`p \\eq 0.4`

The statistic is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

Replacing the info given we got:

`z=\frac{0.467 -0.4}{\sqrt{(0.4(1-0.4))/(150)}}=1.675`

Now we can calculate th p value using the alternative hypothesis with the following probability:

`p_v =2*P(z>1.675)=0.0939`

Since the p value is higher than the significance level given of 0.05 we have enough evidence to FAIL to reject the null hypothesis and we can conclude that the true proportion of Type A donations is not significantly different from 0.4 or 40% at 5% of significance.