NeverReady batteries has engineered a newer, longer lasting AAA battery. The company claims this battery has an average life span of $17$ hours with a standard deviation of $0.8$ hours. Your statistics class questions this claim. As a class, you randomly select $30$ batteries and find that the sample mean life span is $16.7$ hours. If the process is working properly, what is the probability of getting a random sample of $30$ batteries in which the sample mean lifetime is $16.7$ hours or less? Is the company’s claim reasonable?

Given in the question that, NeverReady batteries has engineered a newer, longer lasting AAA battery. The company claims this battery has an average life span of$17$hours with a standard deviation of $0.8$hours. Your statistics class questions this claim. As a class, you randomly select 30 batteries and find that the sample mean life span ishours.

According to the supplied facts, the average life span of a battery is $\mu =17$hours, standard deviation $\sigma =0.8$hours, the sample mean is $\overline{x}=16.7$, and the sample size is $30$.

So, the sampling distribution of sample mean is,

$\overline{X}~N\left({\mu }_x,\frac{{\sigma }_x}{\sqrt{n}}\right)$

$\overline{X}-N\left(17,\frac{0.8}{\sqrt{30}}\right)$

$\overline{X}~N(17,0.146)$

Now, the required probability is $P(\overline{X}\le 16.7)$.

The Ti-83 calculator is used to calculate the probability of acquiring a random sample of $30$batteries with a sample mean lifetime of $16.7$hours or less. To do so, go to $2^{\text{nd}}$, then DISTR, and then scroll down to the normalcdf option and fill in the provided information. After that, press the calculator's ENTER button to get the desired result. The following is a screenshot: