SOCR EduMaterials Activities Normal Distributions

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This is an activity to explore some continuous distributions and the normal approximation to binomial and Poisson.


  • Exercise 1: \item[a.] Graph and print  X \sim exp(0.2) ,  X \sim exp(1) ,  X \sim exp(10)
    • 1. Locate the maximum density for each one of these distributions.
    • 2. Find the height of the density at 3 values of X (one near 0, one near the mean, and one towards the tail of the distribution).
    • 3. Find one percentile for each of these distributions and record them on the printouts. Verify these percentiles using the formula we discussed in class:


x_p=\frac{ln(1-\frac{p}{100})}{-\lambda}

  • 4. Compute one cumulative probability for each one of these distributions, show it on the graph, and verify it with the formula:


P(X \le x)=1-e^{-\lambda x}

  • Exercise 2: Use SOCR to graph and print the distribution of  X \sim N(20, 3) . Show on the graph the following points: \mu \pm 1 \sigma, \mu \pm 2 \sigma, \mu \pm 3 \sigma . How many standard deviations from the mean is the value x = 27.5?
  • Exercise 3: Graph the distribution of  X \sim N(40, 10) .
    • 1. Find P(X > 49) Submit a printout.
    • 2. Find P(X < 22) Submit a printout.
    • 3. Find P(X < 58) Submit a printout.
    • 4. Find P(X > 13) Submit a printout.
    • 5. Find P(12 < X < 37) Submit a printout.
    • 6. Find P(33 < X < 60) Submit a printout.
    • 7. Find P(52 < X < 65) Submit a printout.
    • 8. Use the mouse or the left cut off or right cut off points to find the 8th,20th,45th,55th$,$70th,95th percentiles. After you find these percentiles you can place by hand all of them in one printout (or you can submit a printout for each one of them if you want).
    • 9. Make sure you know how to answer the above questions using the z score z=\frac{x-\mu}{\sigma} and your z table from the handout! You do not need to submit anything here.
  • Exercise 3: The lifetime of tires of brand A follows the normal distribution with mean 40000 miles and standard deviation 4000 miles.
    • 1. Use SOCR to find the probability that a tire will last between 40000 and 46000 miles.
    • 2. Given that a tire will last more than 46000 miles what is the probability that it will last more than 50000 miles? Submit a printout and explain how you get the answer.
    • 3. Given that a tire will last more than 46000 miles what is the probability that it will last less than 50000 miles? Submit a printout and explain how you get the answer.
  • Exercise 4: The probability that a student is admitted in the Math Department Major at a college is <math.30 \%</math>. Suppose that this year 150 students will apply for admission into the Math major.
    • 1. What is the distribution of the number of students admitted? Use $SOCR$ to graph and print this distribution. What is the shape of this distribution? What is the mean and standard deviation of this distribution?
    • 2. Write an expression for the exact probability that among the 150 students at least 55 will be admitted.
    • 3. Use SOCR to compute the probability of part (2).
    • 4. Use the normal distribution to approximate the probability of part (2) (do not forget the continuity correction). What is the error of the approximation?
  • Exercise 5: The number of pine trees per acre in a forest follows the Poisson distribution with mean 30.
    • 1. Graph and print this distribution.
    • 2. What is the shape of this distribution? Why?
    • 3. Use SOCR to find the probability that an acre has more than 37 pines.
    • 4. Approximate the probability of part (c) using the normal distribution.


Below you can see the distribution of a normal random variable X with  \mu=50, \ \sigma=5 . In this graph you can also see the probability that X is between 53 and 60.






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