Question

1. the average life a manufacturers blender is 5 years, with a standard deviation of 1 year. assuming that the lives of these blenders follow approximately a normal distribution, find the probability that the mean life a random sample of 25 such blenders falls between 4.5 and 5.2 years. a) 0.1587 b) 0.8413 c) 0.8351 d) 0.2707 e) 0.1649 f) none of the above.

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 1$ and $n = 25$, then $\sigma_{\bar{x}}=\frac{1}{\sqrt{25}}=\frac{1}{5}=0.2$.

Step2: Calculate the z - scores

The z - score formula is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$.
For $\bar{x}=4.5$, $z_1=\frac{4.5 - 5}{0.2}=\frac{-0.5}{0.2}=-2.5$.
For $\bar{x}=5.2$, $z_2=\frac{5.2 - 5}{0.2}=\frac{0.2}{0.2}=1$.

Step3: Find the probabilities using the standard normal table

From the standard normal table, $P(Z < - 2.5)=0.0062$ and $P(Z < 1)=0.8413$.
The probability $P(-2.5So $P(-2.5

Answer:

C. 0.8351