Answer to Question #120454 in Statistics and Probability for aryan

Question

The price of a popular tennis racket at a national chain store is $179. Portia bought ten of the same racket at an online auction site for the following prices: 155, 179, 175, 175, 161, 158, 170, 165, 163 and 172. Assuming that the auction prices of rackets are normally distributed, determine whether there is sufficient evidence in the sample, at the 5% level of significance, to conclude that the average price of the racket is less than $179 if purchased at an online auction.

Answer 1

$n=10,\\ \bar x=\frac{155+179+...+172}{10}=167.3,\\ s^{2}=\frac{(155-167.3)^{2}+(179-167.3)^{2}+...(172-167.3)^{2}}{9}=65.1222\\ s=\sqrt{65.122}=8.07\\ H_0: \mu\geq179\\ H_1:\mu <179\\ \alpha =0.05\implies t_{(0.05,9)}=1.833,\\ T=\frac{\bar x-\mu_0}{\frac{s}{\sqrt n}}\\ T=\frac{167.3-179}{\frac{8.07}{\sqrt {10}}}=-4.585\\ \text{the rejection region} :\; t<-1.833\\ \text{The decision is to reject}:\; H_0\\ \text {This mean that there is sufficient evidence to }\\ \text{ conclude that the average price of the racket is }\\ less \;than\; 179$