Answer :
Let's work through each of the parts of the problem step-by-step:
### Part (a): Formulate the Hypothesis Test
First, we need to determine the hypothesis to test whether there has been an increase in the proportion of rooms occupied over the one-year period.
- Let [tex]\( p_1 \)[/tex] be the population proportion of rooms occupied for the current year.
- Let [tex]\( p_2 \)[/tex] be the population proportion of rooms occupied for the previous year.
The hypotheses are:
[tex]\[ \begin{aligned} H_0: p_1 - p_2 \leq 0 & \quad \text{(no increase or a decrease in the proportion of rooms occupied)} \\ H_a: p_1 - p_2 > 0 & \quad \text{(an increase in the proportion of rooms occupied)} \end{aligned} \][/tex]
### Part (b): Estimated Proportion of Hotel Rooms Occupied Each Year
We need to calculate the estimated proportions:
Current Year:
The number of occupied rooms is 1,435 out of 1,750 total rooms.
[tex]\[ \hat{p}_1 = \frac{1,435}{1,750} = 0.82 \][/tex]
Previous Year:
The number of occupied rooms is 1,501 out of 1,900 total rooms.
[tex]\[ \hat{p}_2 = \frac{1,501}{1,900} = 0.79 \][/tex]
### Part (c): Conduct the Hypothesis Test
Step 1: Calculate the pooled proportion ([tex]\( \hat{p} \)[/tex])
The pooled proportion is calculated as:
[tex]\[ \hat{p} = \frac{\text{occupied rooms in current year} + \text{occupied rooms in previous year}}{\text{total rooms in current year} + \text{total rooms in previous year}} \][/tex]
[tex]\[ \hat{p} = \frac{1,435 + 1,501}{1,750 + 1,900} \][/tex]
[tex]\[ \hat{p} = \frac{2,936}{3,650} \][/tex]
[tex]\[ \hat{p} = 0.8041 \][/tex]
Step 2: Calculate the standard error
The standard error for the difference in proportions is given by:
[tex]\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
[tex]\[ SE = \sqrt{0.8041 \cdot (1 - 0.8041) \left( \frac{1}{1,750} + \frac{1}{1,900} \right)} \][/tex]
[tex]\[ SE \approx 0.0132 \][/tex]
Step 3: Calculate the test statistic
The test statistic (z) is given by:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
[tex]\[ z = \frac{0.82 - 0.79}{0.0132} \][/tex]
[tex]\[ z \approx 2.27 \][/tex]
Step 4: Determine the p-value
Using the standard normal distribution table, we find the p-value corresponding to [tex]\( z \approx 2.27 \)[/tex]:
[tex]\[ \text{p-value} \approx 0.0116 \][/tex]
Decision Rule:
At a 0.05 significance level ([tex]\(\alpha = 0.05\)[/tex]):
- If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis [tex]\(H_0\)[/tex].
Since [tex]\( 0.0116 < 0.05 \)[/tex], we reject the null hypothesis.
Conclusion:
We conclude that there has been an increase in the hotel occupancy rate.
### Part (d): 95% Confidence Interval Estimate of the Change in Occupancy
To find the confidence interval for the difference in proportions:
The margin of error (ME) for the 95% confidence interval is calculated using:
[tex]\[ ME = z^ \cdot SE \][/tex]
For a 95% confidence interval, [tex]\( z^ = 1.96 \)[/tex]:
[tex]\[ ME = 1.96 \cdot 0.0132 \approx 0.0258 \][/tex]
The confidence interval is given by:
[tex]\[ (\hat{p}_1 - \hat{p}_2) \pm ME \][/tex]
Plugging in the values:
[tex]\[ (0.82 - 0.79) \pm 0.0258 \][/tex]
[tex]\[ 0.03 \pm 0.0258 \][/tex]
[tex]\[ (0.0042, 0.0558) \][/tex]
So, the 95% confidence interval estimate of the change in occupancy for the one-year period is approximately (0.0042, 0.0558).
Thus, we have provided a detailed step-by-step solution to the given problem, including formulating the hypothesis test, calculating proportions, conducting the hypothesis test, and determining the confidence interval.
### Part (a): Formulate the Hypothesis Test
First, we need to determine the hypothesis to test whether there has been an increase in the proportion of rooms occupied over the one-year period.
- Let [tex]\( p_1 \)[/tex] be the population proportion of rooms occupied for the current year.
- Let [tex]\( p_2 \)[/tex] be the population proportion of rooms occupied for the previous year.
The hypotheses are:
[tex]\[ \begin{aligned} H_0: p_1 - p_2 \leq 0 & \quad \text{(no increase or a decrease in the proportion of rooms occupied)} \\ H_a: p_1 - p_2 > 0 & \quad \text{(an increase in the proportion of rooms occupied)} \end{aligned} \][/tex]
### Part (b): Estimated Proportion of Hotel Rooms Occupied Each Year
We need to calculate the estimated proportions:
Current Year:
The number of occupied rooms is 1,435 out of 1,750 total rooms.
[tex]\[ \hat{p}_1 = \frac{1,435}{1,750} = 0.82 \][/tex]
Previous Year:
The number of occupied rooms is 1,501 out of 1,900 total rooms.
[tex]\[ \hat{p}_2 = \frac{1,501}{1,900} = 0.79 \][/tex]
### Part (c): Conduct the Hypothesis Test
Step 1: Calculate the pooled proportion ([tex]\( \hat{p} \)[/tex])
The pooled proportion is calculated as:
[tex]\[ \hat{p} = \frac{\text{occupied rooms in current year} + \text{occupied rooms in previous year}}{\text{total rooms in current year} + \text{total rooms in previous year}} \][/tex]
[tex]\[ \hat{p} = \frac{1,435 + 1,501}{1,750 + 1,900} \][/tex]
[tex]\[ \hat{p} = \frac{2,936}{3,650} \][/tex]
[tex]\[ \hat{p} = 0.8041 \][/tex]
Step 2: Calculate the standard error
The standard error for the difference in proportions is given by:
[tex]\[ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
[tex]\[ SE = \sqrt{0.8041 \cdot (1 - 0.8041) \left( \frac{1}{1,750} + \frac{1}{1,900} \right)} \][/tex]
[tex]\[ SE \approx 0.0132 \][/tex]
Step 3: Calculate the test statistic
The test statistic (z) is given by:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
[tex]\[ z = \frac{0.82 - 0.79}{0.0132} \][/tex]
[tex]\[ z \approx 2.27 \][/tex]
Step 4: Determine the p-value
Using the standard normal distribution table, we find the p-value corresponding to [tex]\( z \approx 2.27 \)[/tex]:
[tex]\[ \text{p-value} \approx 0.0116 \][/tex]
Decision Rule:
At a 0.05 significance level ([tex]\(\alpha = 0.05\)[/tex]):
- If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis [tex]\(H_0\)[/tex].
Since [tex]\( 0.0116 < 0.05 \)[/tex], we reject the null hypothesis.
Conclusion:
We conclude that there has been an increase in the hotel occupancy rate.
### Part (d): 95% Confidence Interval Estimate of the Change in Occupancy
To find the confidence interval for the difference in proportions:
The margin of error (ME) for the 95% confidence interval is calculated using:
[tex]\[ ME = z^ \cdot SE \][/tex]
For a 95% confidence interval, [tex]\( z^ = 1.96 \)[/tex]:
[tex]\[ ME = 1.96 \cdot 0.0132 \approx 0.0258 \][/tex]
The confidence interval is given by:
[tex]\[ (\hat{p}_1 - \hat{p}_2) \pm ME \][/tex]
Plugging in the values:
[tex]\[ (0.82 - 0.79) \pm 0.0258 \][/tex]
[tex]\[ 0.03 \pm 0.0258 \][/tex]
[tex]\[ (0.0042, 0.0558) \][/tex]
So, the 95% confidence interval estimate of the change in occupancy for the one-year period is approximately (0.0042, 0.0558).
Thus, we have provided a detailed step-by-step solution to the given problem, including formulating the hypothesis test, calculating proportions, conducting the hypothesis test, and determining the confidence interval.
