Question

A random sample of 144 recent donations at a certain blood bank reveals that 81 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 the appropriate hypotheses using a significance level of 0.01. State the appropriate null and alternative hypotheses.

Answer 1

Null hypothesis:
`p=0.4`

Alternative hypothesis:
`p \\eq 0.4`

`z=\frac{0.5625 -0.4}{\sqrt{(0.4(1-0.4))/(144)}}=3.98`

`p_v =2*P(z>3.98)=0.0000689`

Since the p value is very low compared to the significance level we have enough evidence to reject the null hypothesis and we can conclude that the true percent of people with type A of blood is significantly different from 0.4 or 40%

Explanation:

Information given

n=144 represent the random sample taken

X=81 represent the number of people with type A blood

`\hat p=(81)/(144)=0.5625` estimated proportion of people with type A blood

`p_o=0.4` is the value that we want to verify

`\alpha=0.01` represent the significance level

z would represent the statistic

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{0.5625 -0.4}{\sqrt{(0.4(1-0.4))/(144)}}=3.98`

Now we can calculate the p value with this probability taking in count the alternative hypothesis:

`p_v =2*P(z>3.98)=0.0000689`

Since the p value is very low compared to the significance level we have enough evidence to reject the null hypothesis and we can conclude that the true percent of people with type A of blood is significantly different from 0.4 or 40%