Question

It is known that cellular phones batteries by a certain factory have lifetimes that follow a normal distribution. The average lifetime of the batteries is known to be 2.4 years. A random sample size of 16 will be taken.

1. Determine the mean and the standard error of the sampling distribution of the means.

2. What is the distribution of the sampling distribution of the sample mean?

3. What is the probability that a random sample size of 16 will have a sample mean greater than 2.5 years?

4. What is the probability that a random sample of size 16 will have a sample mean of less than 2.2 years?

Answer 1

Let $\bar{X}=$ the sample mean.

1.

$$\mu_{\bar{X}}=\mu=2.4\ years$$

$$\sigma_{\bar{X}}=\sqrt{\sigma_{\bar{X}}^2}=\sqrt{\sigma^2/n}$$

$$=\sqrt{0.32^2/16}=0.08\ years$$

2.

$$\bar{X}\sim N(2.4, 0.08^2)$$

3.

$$P(\bar{X}>2.5)=1-P(\bar{X}\le 2.5)$$

$$=1-P(Z\le \dfrac{2.5-2.4}{0.08})$$

$$=1-P(Z\le1.25)\approx0.105650$$

4.

$$P(X<2.2)=P(Z\le \dfrac{2.2-2.4}{0.08})$$

$$=P(Z<-2.5)\approx0.006210$$