1. ## CLT question

A soil scientist wants estimate the average pH for a large field by randomly selecting 40 core samples. Although the population standard deviation of pH is not know, past experiences indicates that most soils have a pH value between 5 and 8. Find the approximate probability that the sample mean will be within .2 units of the true average pH.

$P\left( -\frac{0.2}{\sigma/\sqrt{40}} \leq \frac{\overline{Y}-\mu}{\sigma/\sqrt{n}} \leq \frac{0.2}{\sigma/\sqrt{40}}\right)$

$P\left( -\frac{0.2}{\sigma/\sqrt{40}} \leq Z \leq \frac{0.2}{\sigma/\sqrt{40}}\right)$

I'm just have trouble deciding on a $\sigma$, since it's between 5 and 8, I'm not sure if I should just take an average which is 6.5, or if there's some other way of getting it.

2. Originally Posted by lllll
A soil scientist wants estimate the average pH for a large field by randomly selecting 40 core samples. Although the population standard deviation of pH is not know, past experiences indicates that most soils have a pH value between 5 and 8. Find the approximate probability that the sample mean will be within .2 units of the true average pH.

$P\left( -\frac{0.2}{\sigma/\sqrt{40}} \leq \frac{\overline{Y}-\mu}{\sigma/\sqrt{n}} \leq \frac{0.2}{\sigma/\sqrt{40}}\right)$

$P\left( -\frac{0.2}{\sigma/\sqrt{40}} \leq Z \leq \frac{0.2}{\sigma/\sqrt{40}}\right)$

I'm just have trouble deciding on a $\sigma$, since it's between 5 and 8, I'm not sure if I should just take an average which is 6.5, or if there's some other way of getting it.
"past experiences indicates that most soils have a pH value between 5 and 8." does not mean $5 < \sigma < 8$. It means that 5 < Y < 8, more or less.

Perhaps you could use the fact that for many distributions $\Pr(\mu - 3 \sigma < Y < \mu + 3 \sigma) \approx 1$. Then an estimate for $\sigma$ could be found by solving $\mu - 3 \sigma = 5$ and $\mu + 3 \sigma = 8$ for $\sigma$.

A competent second opinion might confirm or deny this suggestion .....