Stat 225

Continuous Quiz Review

We’ll have quiz this Thursday, March 7. You may use a calculator and you can expect that you’ll have a normal table like this one. There are a few practice problems below and we’ll practice more in class together on Tuesday.

  1. Let \[f_{\lambda}(x) = \begin{cases} \lambda e^{-3x} & x \geq 0 \\ 0 & \text{else}. \end{cases} \]

    1. Find the value of \(\lambda\) that makes \(f_{\lambda}\) a valid probability distribution.
    2. Assuming \(X\) has this distribution, compute \(P(X<0.5)\)

  2. Suppose that \(X\) is normally distributed with mean \(65\) and standard deviation \(8\). Find \(P(62<X<70)\).

  3. Suppose my class’s exam scores are normally distributed with a mean of \(60\) and a standard deviation of \(15\). What percentage of my students score above 90%?

  4. Use \(u\)-substitution to translate the normal integral \[\frac{1}{\sqrt{200\pi}} \int_{100}^{115} e^{-(x-110)^2/200}dx\] into a standard normal integral.

  5. A sample of 123 North Carolina men finds their average weight to be 190 pounds with a standard deviation of 36.75 pounds.

    1. What is the standard error associated with this sample mean?

    2. Write down a 96% confidence interval for the weight of a North Carolina man based on this sample.

  6. A sample of 12 North Carolinians finds that 5 of them have smoked a significant amount.

    1. What is the standard error associated with this sample proportion?

    2. Write down a 95% confidence interval for the height of a North Carolina adult based on this sample.