Answer to Question #280327 in Statistics and Probability for Sipu

Question

The average hospitality girls working in a certain area is 19 years old. A civic oriented group made a recent survey on the age of the hospitality girls working in the same locality. A random sample of 20 respondents showed an average of 15.4 years old with a standard deviation of 2.14 years old. Using a 0.05 level of significance, has the hospitality girls gone lower?

Answer 1

Let $X$ be a random variable representing the age of hospitality girls.

$n=20,\space \bar{x}=15.4,\space s=2.14$

The hypotheses tested are,

$H_0:\mu=19 \space vs\space H_1:\mu\lt 19$

The test statistic is given as,

$t={(\bar{x}-\mu)\over{s\over\sqrt{n}}}={15.4-19\over {2.14\over\sqrt{20}}}={-3.6\over0.4785}=-7.5232$

$t$ is compared with the t distribution table value at $\alpha=0.05$ with $n-1=20-1=19$ degrees of freedom.

The table value is given as,

$t_{0.05,19}=-1.729133$.

The null hypothesis is rejected if $t\lt t_{0.05,19}$

Since $t=-7.5232\lt t_{0.05,19}=-1.729133$, we reject the null hypothesis and conclude that there is sufficient evidence to show that the age of hospitality girls has gone lower at 5% level of significance.