Answer:
Yes, we have reason to believe that fewer than one-fifth are heated by oil.
Step-by-step explanation:
A one-sample proportion test is to be performed to determine whether fewer than one-fifth of the homes in a certain city are heated by oil.
The hypothesis can be defined as follows:
<em>H</em>₀: The proportion of homes in a certain city that are heated by oil is not less than one-fifth, i.e. <em>p</em> ≥ 0.20.
<em>H</em>ₐ: The proportion of homes in a certain city that are heated by oil is less than one-fifth, i.e. <em>p</em> < 0.20.
The information provided is:
<em>n</em> = 1000
<em>x</em> = 136
<em>α</em> = 0.05
Compute the sample proportion as follows:
![\hat p=\frac{x}{n}=\frac{136}{1000}=0.136](https://tex.z-dn.net/?f=%5Chat%20p%3D%5Cfrac%7Bx%7D%7Bn%7D%3D%5Cfrac%7B136%7D%7B1000%7D%3D0.136)
Compute the test statistic as follows:
![z=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B%5Chat%20p-p%7D%7B%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%7D%7D)
![=\frac{0.136-0.20}{\sqrt{\frac{0.136(1-0.136)}{1000}}}\\\\=-5.9041\\\\\approx -5.90](https://tex.z-dn.net/?f=%3D%5Cfrac%7B0.136-0.20%7D%7B%5Csqrt%7B%5Cfrac%7B0.136%281-0.136%29%7D%7B1000%7D%7D%7D%5C%5C%5C%5C%3D-5.9041%5C%5C%5C%5C%5Capprox%20-5.90)
The test statistic value is, -5.90.
Decision rule:
Reject the null hypothesis if the <em>p</em>-value of the test is less than the significance level.
Compute the <em>p</em>-value as follows:
![p-value=P(Z](https://tex.z-dn.net/?f=p-value%3DP%28Z%3C-5.90%29%5C%5C%5C%5C%3D1-P%28Z%3C5.90%29%5C%5C%5C%5C%3D1-%28%5Capprox%201%29%5C%5C%5C%5C%3D0)
The <em>p-</em>value of the test is, 0.
<em>p</em>-value = 0 < <em>α</em> = 0.05
The null hypothesis will be rejected at 5% level of significance.
Conclusion:
The proportion of homes in a certain city that are heated by oil is less than one-fifth.