Answer to Question #291884 in Statistics and Probability for Belete Dessalegn

Question

1.     The mean life time of a sample of 16 fluorescent light bulbs produced by a company is computed to be 1570 hours. The population standard deviation is 120 hours. Suppose the hypothesized value for the population mean is 1600 hours. Can we conclude that the life time of light bulbs is decreasing?   (Use  and assume the normality of the population) 

Answer 1

$n=16\\\sigma=120\\\bar{x}=1570$

The hypotheses tested are,

$H_0:\mu=1600\\vs\\H_1:\mu\lt1600$

The test statistic is given as,

$Z_c= {\bar x-\mu\over{\sigma\over\sqrt{n}}}={1570-1600\over{120\over\sqrt {16}}}={-30\over30}=-1$

The critical value at $\alpha=0.05$ is,

$Z_{0.05}=1.645$

The null hypothesis is rejected if $|Z_c|\gt Z_{0.05}$

Here, $|Z_c|=1\lt Z_{0.05}=1.645$. Therefore, we fail to reject the null hypothesis and conclude that there is no sufficient evidence to show that the life time of light bulbs is decreasing at 5% level of significance.