Answer:
Step-by-step explanation:
Given that:
![H_o: \mu \ge 55 \ \\ \\ H_1 : \mu < 55](https://tex.z-dn.net/?f=H_o%3A%20%5Cmu%20%5Cge%2055%20%5C%20%5C%5C%20%5C%5C%20H_1%20%3A%20%5Cmu%20%3C%2055)
(a) For x = 54 and s = 5.3
The test statistics can be computed as:
![Z = \dfrac{\overline x - \mu}{\dfrac{s}{\sqrt{n}} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B%5Coverline%20x%20-%20%5Cmu%7D%7B%5Cdfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%20%7D)
![Z = \dfrac{54-55}{\dfrac{5.3}{\sqrt{36}} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B54-55%7D%7B%5Cdfrac%7B5.3%7D%7B%5Csqrt%7B36%7D%7D%20%7D)
![Z = \dfrac{-1}{\dfrac{5.3}{6} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B-1%7D%7B%5Cdfrac%7B5.3%7D%7B6%7D%20%7D)
Z = -1.132
degree of freedom = n - 1
= 36 - 1
= 35
Using the Excel Formula:
P-Value = T.DIST(-1.132,35,1) = 0.1326
Decision: p-value is greater than significance level; do not reject ![\mathbf{H_o}](https://tex.z-dn.net/?f=%5Cmathbf%7BH_o%7D)
![\mu < 55](https://tex.z-dn.net/?f=%5Cmu%20%3C%2055)
b
For x = 53 and s = 4.6
The test statistics can be computed as:
![Z = \dfrac{\overline x - \mu}{\dfrac{s}{\sqrt{n}} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B%5Coverline%20x%20-%20%5Cmu%7D%7B%5Cdfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%20%7D)
![Z = \dfrac{53-55}{\dfrac{4.6}{\sqrt{36}} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B53-55%7D%7B%5Cdfrac%7B4.6%7D%7B%5Csqrt%7B36%7D%7D%20%7D)
![Z = \dfrac{-2}{\dfrac{4.6}{6} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B-2%7D%7B%5Cdfrac%7B4.6%7D%7B6%7D%20%7D)
Z = -2.6087
degree of freedom = n - 1
= 36 - 1
= 35
Using the Excel Formula:
P-Value = T.DIST(-2.6087,35,1) =0.0066
Decision: p-value is < significance level; we reject the null hypothesis.
![\mu < 55](https://tex.z-dn.net/?f=%5Cmu%20%3C%2055)
c)
For x = 56 and s = 5.0
The test statistics can be computed as:
![Z = \dfrac{\overline x - \mu}{\dfrac{s}{\sqrt{n}} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B%5Coverline%20x%20-%20%5Cmu%7D%7B%5Cdfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%20%7D)
![Z = \dfrac{56-55}{\dfrac{5.0}{\sqrt{36}} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B56-55%7D%7B%5Cdfrac%7B5.0%7D%7B%5Csqrt%7B36%7D%7D%20%7D)
![Z = \dfrac{56-55}{\dfrac{5.0}{\sqrt{36}} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7B56-55%7D%7B%5Cdfrac%7B5.0%7D%7B%5Csqrt%7B36%7D%7D%20%7D)
Z = 1.2
degree of freedom = n - 1
= 36 - 1
= 35
Using the Excel Formula:
P-Value = T.DIST(1.2,35,1) = 0.88009
Decision: p-value is greater than significance level; do not reject ![\mathbf{H_o}](https://tex.z-dn.net/?f=%5Cmathbf%7BH_o%7D)
![\mu < 55](https://tex.z-dn.net/?f=%5Cmu%20%3C%2055)