Question

A random sample of 150 recent donations at a certain blood bank reveals that 82 were type A blood. Does this suggest that the actual percentage of type A donations differs from 40%, the percentage of 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

Concepts and reason

Sampling Distribution of Proportion:

The random sample of size n is taken from the population with sample proportion $\hat p$ .

The sampling distribution of the sample proportion has mean $p$ and the standard deviation $\sqrt {\frac{{p\left( {1 - p} \right)}}{n}}$ . Moreover, the sample proportion follows normal distribution for large sample size n.

The null and alternative hypothesis for proportion test is as shown below:

The null hypothesis is denoted as ${H_0}$ and defined as,

$\begin{array}{l}\\{H_0}:p = {p_0}\\\\{H_0}:p \le {p_0}\\\\{H_0}:p \ge {p_0}\\\end{array}$

Hence, $p$ be the population proportion and ${p_0}$ is the hypothesized value.

The alternative hypothesis is denoted as ${H_1}$ and defined as,

$\begin{array}{l}\\{H_a}:p \ne {p_0}\left( {{\rm{Two tailed}}} \right)\\\\{H_a}:p < {p_0}\left( {{\rm{Left tailed}}} \right)\\\\{H_a}:p \ge {p_0}\left( {{\rm{Right tailed}}} \right)\\\end{array}$

Hence, $p$ be the population proportion and ${p_0}$ is the hypothesized value.

Fundamentals

The formula for sample proportion is,

$\hat p\,\, = \,\,\frac{X}{n}$

Formula for z-score of sample proportion is given below:

$z = \frac{{\hat p - {p_0}}}{{\sqrt {\frac{{{p_0}\left( {1 - {p_0}} \right)}}{n}} }}$

Where p is the population proportion and $\hat p$ is the sample proportion.

Rejection region (two tailed):

Reject the null hypothesis if $z \ge {z_{\alpha /2}}{\rm{ or }}z \le - {z_{\alpha /2}}$ .

Rejection region (right tailed):

Reject the null hypothesis if $z \ge {z_{\alpha /2}}$ .

Rejection region (left tailed):

Reject the null hypothesis if $z \le - {z_{\alpha /2}}$ .

Let p be the population proportion of type A blood.

The null and alternative hypotheses are,

Null hypothesis: There is no difference in the actual percentage of the population having type A blood is 40 %.

${H_0}:p = \,\,0\cdot4$

Alternative hypothesis: The actual percentage of the population having type A blood differs from 40 %.)

The specified level of significance is, $\alpha \, = \,0.01$ .

From the standard normal distribution table, the critical values at 0.01 level of the significance for two tailed test is $\pm 2.576$

Rejection region: reject the null hypothesis if $z \ge 2.576{\rm{ or }}z \le - 2.576$ .

Here, X is the number of observations, and n is the sample size.

Let the number of persons having a Type A blood be, $X = 82$ .

Let the number of donations be, $n = 150$ .

The sample proportion is,

$\begin{array}{c}\\\hat p\,\, = \,\,\frac{X}{n}\\\\ = \frac{{82}}{{150}}\,\,\\\\ = 0.5466\\\end{array}$

The formula for the test statistic is,

$z = \frac{{\hat p - {p_0}}}{{\sqrt {\frac{{{p_0}\left( {1 - {p_0}} \right)}}{n}} }}$

$\begin{array}{c}\\z = \frac{{0.5466 - 0\cdot4}}{{\sqrt {\frac{{0\cdot4 \times 0\cdot6}}{{150}}} }}\\\\ = 3.6666\\\\ \approx 3.667\\\end{array}$

Since, the test statistic value of 3.667 exceeds the critical value of 2.576. So, reject the null hypothesis. It can be concluded that actual percentage of the population having type A blood differs significantly from 40%.

The specified level of significance is, $\alpha \, = 0.05$ .

From the standard normal distribution table, the critical values at 0.05 level of the significance for two tailed test is $\pm 1.96$

Rejection region: reject the null hypothesis if $z \ge 1.96{\rm{ or }}z \le - 1.96$ .

Since, the test statistic value of 3.667 exceeds the critical value of 1.96. So, reject the null hypothesis. It can be concluded that actual percentage of the population having type A blood differs significantly from 40%.

Ans:

The actual percentage of the population having type A blood differs significantly from 40%.

The actual percentage of the population having type A blood differs significantly from 40%.