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 4.8 and 5.3 years. a) 0.1972 b) 0.2255 c) 0.9332 d) 0.0668 e) 0.7745 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 $\bar{x}_1 = 4.8$, $z_1=\frac{4.8 - 5}{0.2}=\frac{- 0.2}{0.2}=-1$.
For $\bar{x}_2 = 5.3$, $z_2=\frac{5.3 - 5}{0.2}=\frac{0.3}{0.2}=1.5$.

Step3: Find the probabilities

Using the standard normal distribution table, $P(Z\lt - 1)=0.1587$ and $P(Z\lt1.5)=0.9332$.
The probability $P(4.8\lt\bar{X}\lt5.3)=P(Z\lt1.5)-P(Z\lt - 1)$.
Substitute the values: $P(4.8\lt\bar{X}\lt5.3)=0.9332-0.1587 = 0.7745$.

Answer:

E) 0.7745