Question

Tourism is extremely important to the economy of Florida. Hotel occupancy is an often-reported measure of visitor volume and visitor activity (Orlando Sentinel). Hotel occupancy data for February in two consecutive years are as follows.
Current Year Previous Year
Occupied Rooms 1,470 1,458
Total Rooms 1,750 1,800

Required:
a. Formulate the hypothesis test that can be used to determine if there has been an increase in the proportion of rooms occupied over the one-year period.
b. What is the estimated proportion of hotel rooms occupied each year?
c. Calculate the test statistic.
d. What is the p-value?

Answer 1

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:

 $\hat p_{1}=\frac{X_{1}}{n_{1}}=\frac{1470}{1750}=0.84\\\\\hat p_{2}=\frac{X_{2}}{n_{2}}=\frac{1458}{1800}=0.81\\\\\hat p=\frac{X_{1}+X_{2}}{n_{1}+n_{2}}=\frac{1470+1458}{1750+1800}=0.825$

(c)

Compute the test statistic value as follows:

$Z=\frac{\hat p_{1}-\hat p_{2}}{\sqrt{\hat p(1-\hat p)\times [\frac{1}{n_{1}}+\frac{1}{n_{2}}]}}$

   $=\frac{0.84-0.81}{\sqrt{0.825(1-0.825)\times [\frac{1}{1750}+\frac{1}{1800}]}}\\\\=2.352$

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:

 $p-value=P(Z>2.352)=1-P(Z$

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.