Question

the average number of moves a person makes in his or her lifetime is 12 and the standard deviation is 3.4. assume that the sample is taken from a large population and the correction factor can be ignored. use the standard normal distribution table. round the final answers to four decimal places and intermediate = value calculations to two decimal places. part 1 of 3 find the probability that the mean of a sample of 25 people is less than 10. ( p(overline{x}<10)=square )

Explanation:

Step1: Calculate the standard error

The formula for the standard error $\sigma_{\bar{x}}$ is $\frac{\sigma}{\sqrt{n}}$, where $\sigma = 3.4$ and $n = 25$.
$$\sigma_{\bar{x}}=\frac{3.4}{\sqrt{25}}=\frac{3.4}{5}=0.68$$

Step2: Calculate the z - score

The formula for the z - score is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, where $\bar{x}=10$, $\mu = 12$ and $\sigma_{\bar{x}} = 0.68$.
$$z=\frac{10 - 12}{0.68}=\frac{-2}{0.68}\approx - 2.94$$

Step3: Find the probability

We want to find $P(\bar{X}<10)$, which is equivalent to $P(Z < - 2.94)$. Using the standard normal distribution table, $P(Z < - 2.94)=0.0016$

Answer:

$0.0016$