The drag Prevnar is a vaccine meant to prevent meningitis. It is typically administered to infants between 12 and 15 months of age. Of the 710 infants,121 experienced a loss of appetite. Is there significant evidence to conclude the proportion of infants who receive Prevnar and experience of a loss of appetite is different from 0.135, the proportion of children who experience a loss of appetite with competing medications? Use α=0.01 level of significance.

Answer:

The pvalue of the test is 0.0058 < 0.01, which means that there is significant evidence to conclude the proportion of infants who receive Prevnar and experience of a loss of appetite is different from 0.135.

Step-by-step explanation:

Test if the proportion of infants who receive Prevnar and experience of a loss of appetite is different from 0.135

This means that the null hypothesis is:

[tex]H_{0}: p = 0.135[/tex]

And the alternate hypothesis is:

[tex]H_{a}: p \neq 0.135[/tex]

The test statistic is:

[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, [tex]\sigma[/tex] is the standard deviation and n is the size of the sample.

0.135 is tested at the null hypothesis:

This means that [tex]\mu = 0.135, \sigma = \sqrt{0.135*0.865}[/tex]

Of the 710 infants,121 experienced a loss of appetite.

This means that [tex]n = 710, X = \frac{121}{710} = 0.1704[/tex]

Value of the test-statistic:

[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

[tex]z = \frac{0.1704 - 0.135}{\frac{\sqrt{0.135*0.865}}{\sqrt{710}}}[/tex]

[tex]z = 2.76[/tex]

Pvalue of the test and decision:

The pvalue of the test is the pobability that the population proportion differs from the tested proportion by at least 0.1704 - 0.135 = 0.0354, which is P(|z| > 2.76). This probability is 2 multiplied by the pvalue of z = -2.76.

Looking at the z-table, z = -2.76 has a pvalue of 0.0029

2*0.0029 = 0.0058

The pvalue of the test is 0.0058 < 0.01, which means that there is significant evidence to conclude the proportion of infants who receive Prevnar and experience of a loss of appetite is different from 0.135.

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