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ANTONII [103]
3 years ago
8

A museum conducts a survey of its visitors in order to assess the popularity of a device which

Mathematics
1 answer:
Natali5045456 [20]3 years ago
7 0

Answer:

i:

The appropriate null hypothesis is H_0: p \geq 0.2

The appropriate alternative hypothesis is H_1: p < 0.2

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is not sufficient evidence that fewer than 20% of the museum visitors make use of the device, and so, it should not be withdrawn.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The device will be withdrawn if fewer than 20% of all of the museum’s visitors make use of it.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:

H_0: p \geq 0.2

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

H_1: p < 0.2

The test statistic is:

z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}

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

0.2 is tested at the null hypothesis:

This means that \mu = 0.2, \sigma = \sqrt{0.2*0.8} = \sqrt{0.16} = 0.4.

The device will be withdrawn if fewer than 20% of all of the museum’s visitors make use of it.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 n = 100, X = \frac{15}{100} = 0.15

Test statistic:

z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}

z = \frac{0.15 - 0.20}{\frac{0.4}{\sqrt{100}}}

z = -1.25

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is not sufficient evidence that fewer than 20% of the museum visitors make use of the device, and so, it should not be withdrawn.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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