Respuesta :
Answer:
Explained below.
Step-by-step explanation:
In this case we need to determine if there has been an increase in the proportion of rooms occupied over the one-year period.
(a)
The hypothesis can be defined as follows:
H₀: The proportion of rooms occupied over the one-year period has not increased, i.e. p₁ - p₂ ≤ 0.
Hₐ: The proportion of rooms occupied over the one-year period has increased, i.e. p₁ - p₂ > 0.
(b)
The information provided is:
n₁ = 1750
n₂ = 1800
X₁ = 1470
X₂ = 1458
Compute the sample proportions and total proportions as follows:
[tex]\hat p_{1}=\frac{X_{1}}{n_{1}}=\frac{1470}{1750}=0.84\\\\\hat p_{2}=\frac{X_{2}}{n_{2}}=\frac{1458}{1800}=0.81\\\\\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{1470+1458}{1750+1800}=0.825[/tex]
(c)
Compute the test statistic value as follows:
[tex]Z=\frac{\hat p_{1}-\hat p_{2}}{\sqrt{\hat p(1-\hat p)\times [\frac{1}{n_{1}}+\frac{1}{n_{2}}]}}[/tex]
[tex]=\frac{0.84-0.81}{\sqrt{0.825(1-0.825)\times [\frac{1}{1750}+\frac{1}{1800}]}}\\\\=2.352[/tex]
The test statistic value is 2.352.
(d)
The decision rule is:
The null hypothesis will be rejected if the p-value of the test is less than the significance level.
Compute the p-value as follows:
[tex]p-value=P(Z>2.352)=1-P(Z<2.352)=1-0.99061=0.00939[/tex]
The p-value of the test is very small. The null hypothesis will be rejected at any significance level.
Thus, there enough evidence suggesting that there has been an increase in the proportion of rooms occupied over the one-year period.
