Respuesta :
Answer:
i:
The appropriate null hypothesis is [tex]H_0: p \geq 0.2[/tex]
The appropriate alternative hypothesis is [tex]H_1: p < 0.2[/tex]
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is not sufficient evidence that fewer than 20% of the museum visitors make use of the device, and so, it should not be withdrawn.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The device will be withdrawn if fewer than 20% of all of the museum’s visitors make use of it.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:
[tex]H_0: p \geq 0.2[/tex]
At the alternative hypothesis, we test if the proportion is less than 20%, that is:
[tex]H_1: p < 0.2[/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.
0.2 is tested at the null hypothesis:
This means that [tex]\mu = 0.2, \sigma = \sqrt{0.2*0.8} = \sqrt{0.16} = 0.4[/tex].
The device will be withdrawn if fewer than 20% of all of the museum’s visitors make use of it.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 [tex]n = 100, X = \frac{15}{100} = 0.15[/tex]
Test statistic:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \frac{0.15 - 0.20}{\frac{0.4}{\sqrt{100}}}[/tex]
[tex]z = -1.25[/tex]
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is not sufficient evidence that fewer than 20% of the museum visitors make use of the device, and so, it should not be withdrawn.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
