1. Let $Z$ be a standard normal random variable. Find the value of $x$ such that $P(Z > x) = 0.05$.

    $4.66$ $2.33$ $3.28$ $1.64$

  2. Consider a sample $X_1,\ldots,X_{ 16}$ of normally distributed random variables with mean $\mu = 2$ and variance $\sigma^2 = 1$. What is the probability that $1.8 \le \bar{X} \le 2.2$?

    $\Phi( 0.8) - \Phi( -0.8) \approx 0.5763$ $\Phi( 0.4) - \Phi( -0.4) \approx 0.3108$ $\Phi( 0.2) - \Phi( -0.2) \approx 0.1585$ $\Phi( 1.2) - \Phi( -1.2) \approx 0.7699$ $\Phi( 0.05) - \Phi( -0.05) \approx 0.0399$

  3. Let $X$ be a normal random variable with $\mu = 10$ and $\sigma = 5$. Find $P( 20 < X < 30)$.

    $1.0 - 0.9772 = 0.0227$ $0.8413 - 0.6915 = 0.1499$ $0.9032 - 0.7881 = 0.1151$ $1.0 - 0.9993 = 7.0 \times 10^{-4}$

  4. The germination time in days of a newly planted seed has the mean value of $2$ and variance $4$. If the germination times $X_1,\ldots,X_n$ are independent, find the mean and the variance for the average germination time $\bar{X}$ of $n = 36$ seeds.

    The mean is ${{1}\over{2}}$ and the variance is ${{4}\over{25}}$ The mean is ${{1}\over{2}}$ and the variance is $4$ The mean is $2$ and the variance is $4$ The mean is ${{1}\over{2}}$ and the variance is ${{1}\over{9}}$ The mean is $2$ and the variance is ${{1}\over{9}}$ The mean is $2$ and the variance is ${{4}\over{25}}$

  5. The germination time in days of a newly planted seed has the mean value of ${{1}\over{2}}$ and variance ${{1}\over{4}}$. If the germination times $X_1,\ldots,X_n$ are independent, estimate the probability that the average germination time $\bar{X}$ of $n = 25$ seeds is between 0.8 and 1.2 days.

    $\Phi( 6.75) - \Phi( 2.75) \approx 0.003$ $\Phi( 13.5) - \Phi( 5.5) \approx 0.0$ $\Phi( 14.0) - \Phi( 6.0) \approx 0.0$ $\Phi( 7.0) - \Phi( 3.0) \approx 0.0013$


Department of Mathematics
Last modified: 2025-02-04