Question

A random sample of 149 recent donations at a certain blood bank reveals that 76 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 appropriate hypotheses using a significance level of 0.01. Would your conclusion have been different if a significance level of 0.05 has been used?

Answer 1

Answer:

Yes it suggest that the actual percentage of type A donations differs from 40%, the percentage of the population having type A blood.

Well if a significance level of 0.05 is used it will not affect the conclusion

Step-by-step explanation:

From the question we are told that

     The  sample size is   $n = 149$

     The  number that where  type A blood is  k =  76

       The population proportion is   $p = 0.40$

       The  significance level is  $\alpha = 0.01$

Generally the sample proportion is mathematically represented as

      $\r p = \frac{k}{n}$

=>    $\r p = \frac{76}{149}$

=>    $\r p = 0.51$

The  Null hypothesis is   $H_o : p = 0.41$

The  Alternative hypothesis is  $H_a : p \ne 0.40$

Next we obtain the critical value of $\alpha$ from the z-table.The value is  

       $Z_{\alpha } = Z_{0.01} = 1.28$

Generally the test statistics is mathematically evaluated as

           $t = \frac{\r p - p }{ \sqrt{ \frac{p(1-p)}{n} } }$    

substituting values

           $t = \frac{0.51 - 0.40 }{ \sqrt{ \frac{0.40 (1-0.40 )}{149} } }$    

           $t =2.74$

So looking at the values for t  and  $Z_{0.01}$ we see that  $t > Z_{0.01}$ so we reject the null hypothesis. Which means that there is no sufficient evidence to support the claim

Now if $\alpha = 0.05$ , the from the z-table the critical value for $\alpha = 0.05$ is  $Z_{0.05} = 1.645$

So comparing the value of  t and  $Z_{0.05} = 1.645$  we see that $t > Z_{0.05}$ hence the conclusion would not be different.