A museum conducts a survey of its visitors in order to assess the popularity of a device which is used to provide information on the museum exhibits. The device will be withdrawn if fewer
than 20% of all of the museum’s visitors make use of it. Of a random sample of 100 visitors,
15 chose to use the device.
i. Carry out an appropriate hypothesis test at the 5% significance level to
see if the device should be withdrawn and state your conclusions.
ii. Calculate the p-value of the test

i:

The appropriate null hypothesis is

The appropriate alternative hypothesis is

The p-value of the test is 0.1057 > 0.05, which means that there is not sufficient evidence that fewer than 20% of the museum visitors make use of the device, and so, it should not be withdrawn.

ii:

The p-value of the test is 0.1057

Step-by-step explanation:

Question i:

The device will be withdrawn if fewer than 20% of all of the museum’s visitors make use of it.

At the null hypothesis, we test if the proportion is of at least 20%, that is:

At the alternative hypothesis, we test if the proportion is less than 20%, that is:

The test statistic is:

In which X is the sample mean, is the value tested at the null hypothesis, is the standard deviation and n is the size of the sample.

0.2 is tested at the null hypothesis:

This means that .

The device will be withdrawn if fewer than 20% of all of the museum’s visitors make use of it. Of a random sample of 100 visitors, 15 chose to use the device.

This means that

Test statistic:

P-value of the test and decision:

The p-value of the test is the probability of finding a sample proportion below 0.15, which is the p-value of z = -1.25.

Looking at the z-table, z = -1.25 has a p-value of 0.1057.

The p-value of the test is 0.1057 > 0.05, which means that there is not sufficient evidence that fewer than 20% of the museum visitors make use of the device, and so, it should not be withdrawn.

Question ii:

The p-value of the test is 0.1057

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