Respuesta :
Answer:
(a) H₀: μ = 5 vs. Hₐ: μ ≠ 5.
(b) The test statistic value is -3.67.
(c) The p-value of the test is 0.0025.
(d) The mean amount of time spent in the shower by an adult is different from 5 minutes.
Step-by-step explanation:
In this case we need to test whether the mean amount of time that college students spend in the shower is significantly different from 5 minutes.
The information provided is:
[tex]n=15\\\bar x=4.29\\s=0.75\\\alpha =0.01[/tex]
(a)
The hypothesis for the test can be defined as follows:
H₀: The mean amount of time spent in the shower by an adult is 5 minutes, i.e. μ = 5.
Hₐ: The mean amount of time spent in the shower by an adult is different from 5 minutes, i.e. μ ≠ 5.
(b)
As the population standard deviation is not known we will use a t-test for single mean.
Compute the test statistic value as follows:
[tex]t=\frac{\bar x-\mu}{s/\sqrt{n}}\\\\=\frac{4.29-5}{0.75/\sqrt{15}}\\\\=-3.67[/tex]
Thus, the test statistic value is -3.67.
(c)
Compute the p-value of the test as follows:
[tex]p-value=2\times P(t_{\alpha/2, (n-1)}<-3.67)\\ =2\times P(t_{0.005, 14}<-3.67)\\=0.0025[/tex]
*Use a t-table.
Thus, the p-value of the test is 0.0025.
(d)
Decision rule:
If the p-value of the test is less than the significance level then the null hypothesis will be rejected and vice-versa.
p-value = 0.0025 < α = 0.01
The null hypothesis will be rejected at 1% level of significance.
Thus, concluding that the mean amount of time spent in the shower by an adult is different from 5 minutes.
