Question: The level of nitrogen oxides in the exhaust of a particular car model varies with mean $0.9$ grams per mile and standard deviation $0.17$ grams per mile.
a) What sample size is needed so that the standard deviation of the sampling distribution of the sample mean ($\sigma_{\bar{x}}$) is $0.01$ grams per mile?
b) If a smaller sample is considered, the standard deviation for $\bar{x}$ ($\sigma_{\bar{x}}$) would be ____________ .
1. **State the problem:**
We know the population standard deviation $\sigma = 0.17$ grams per mile.
We want to find the sample size $n$ such that the standard deviation of the sample mean $\sigma_{\bar{x}} = 0.01$ grams per mile.
2. **Recall the formula for the standard deviation of the sample mean:**
$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$
3. **Set up the equation and solve for $n$: **
$$0.01 = \frac{0.17}{\sqrt{n}}$$
Multiply both sides by $\sqrt{n}$:
$$0.01 \sqrt{n} = 0.17$$
Divide both sides by $0.01$:
$$\sqrt{n} = \frac{0.17}{0.01} = 17$$
Square both sides:
$$n = 17^2 = 289$$
4. **Answer for part (a):**
The required sample size is $\boxed{289}$.
5. **For part (b):**
If a smaller sample size is considered, then $n$ decreases.
Since $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, decreasing $n$ makes $\sqrt{n}$ smaller, so $\sigma_{\bar{x}}$ becomes larger.
6. **Answer for part (b):**
The standard deviation for $\bar{x}$ would be $\boxed{LARGER}$.