Answer to Question #229503 in Statistics and Probability for Robert

Question

A hotel chain is interested to decide wether to convert more of its rooms to family rooms. In a random sample of 500 guests in 1998, 266 had requested family rooms. In 1999, 305 guests in a random sample of 480 preferred the family room. As the hotel manager, you need to recommend wether the hotel chain should convert more rooms to family rooms. Support your recommendation by testing the appropriate hypothesis test at a = 0.05.

Answer 1

$n_1=500 \\ x_1=266 \\ n_2=480 \\ x_2=305$

Level of significance α=0.05

$\hat{p_1}=\frac{x_1}{n_1}= \frac{266}{500}=0.532 \\ \hat{p_2}=\frac{x_2}{n_2}= \frac{305}{480}=0635 \\ \hat{q_1}= 1 -\hat{p_1}= 0.468 \\ \hat{q_2}=1 -\hat{p_2}=0.365 \\ H_0: p_1=p_2 \\ H_1: p_1 <p_2$

Test-statistic:

$Z = \frac{\hat{p_1} -\hat{p_2}}{\sqrt{\frac{\hat{p_1} \hat{q_1}}{n_1} + \frac{\hat{p_2} \hat{q_2}}{n_2} }} \\ = \frac{0.532 -0.635}{\sqrt{\frac{0532 \times 0.468}{500} + \frac{0.635 \times 0.365}{480} }} \\ = \frac{-0.103}{\sqrt{0.000497 +0.000482}} \\ = \frac{-0.103} {0.0312} \\ = -3.3012 \\ One-tail \;test. \\ Z_{α}=Z_{0.05} = -1.96$

$Z<Z_{ α}$ We reject H₀ at α=0.05

The hotel chain should convert more of it’s rooms to family rooms.