Stat 11 Spring 2011 - Homework 7
(1) Which one of the following questions does a hypothesis test address? No explanation is necessary.
(i) Is the difference in the observed and hypothesized values due to chance?
(ii) Is the difference in the observed and hypothesized important?
(iii) What does the difference in the observed and hypothesized values prove?
(iv) Was the experiment properly designed?
(2) Ilsa is an avid bowler. She has an expected score of 180 with an SD of 30. Assume that her scores are Normally distributed and independent of each other. What are the probabilities that:
(a) her score in a game will be above 200?
(b) her score in a game will be below 120?
(c) her mean score for a series of three games will be above 200?
(d) her mean score for a series of three games will be below 120? (answer: .0003)
(e) Suppose Rick is a poor bowler, averaging less than 120 per game. Would Rick have a better chance of beating Ilsa in a single game, or over an entire series of three games? Explain briefly.
(3) The inner planets (Mercury, Venus, Earth, and Mars) are those whose orbits lie inside the asteroid belt; the outer planets (Jupiter, Saturn, Uranus, Neptune) are those whose orbits lie beyond the asteroids. The relative masses of the planets are shown below, with the mass of the Earth taken to be 1.00:
|
Mercury
|
Venus
|
Earth
|
Mars
|
Jupiter
|
Saturn
|
Uranus
|
Neptune
|
|
0.06
|
0.81
|
1.00
|
0.11
|
318
|
95
|
15
|
17
|
The masses of the inner planets average 0.49, while the masses of the outer planets average 111. Does it make sense to carry out a hypothesis test to determine whether this difference is statistically significant? Explain briefly. (Note: Pluto was recently "demoted" and is no longer considered a planet. See https://news.bbc.co.uk/1/hi/world/5282440.stm for details.)
(4) One of the best ways to learn something is to teach it to someone else. Therefore, explain what a p-value is to your roommate, friend, family member, or someone else who hasn't taken a Stat class. You may want to think of an example or analogy to help you explain the concept. It may be helpful to keep in mind that a p-value is a conditional probability. (What's the condition?) There is nothing to hand in.
5. Computer assignment.
Go to the following web site: https://onlinestatbook.com/stat_sim/sampling_dist/. On this page is a Java applet that demonstrates the Central Limit Theorem, which we looked at in class. Wait a few moments for the applet to load; when it is ready the button on the top left of the page will say "Begin". While you're waiting, read the instructions on the web site.
When you click the "Begin" button, a new window with four panels will open. First, check the box for "Fit normal" to the right of the third panel. Now, at the top right of the window, click on the "Clear lower 3" button to reset the applet, then select "Skewed" from the pop-up menu below that button. You should see a skewed histogram in the top panel; this represents the population. Click on "Animated sample" to the right of the second panel. The computer will take a random sample of five values and make a histogram of these values in the second panel. The mean of these five values will be shown in the third panel. Click on "Animated sample" again to take a second random sample of five values, and observe where the second sample mean falls.
You could repeatedly take random samples one at a time in this way, but that would be very time-consuming. Instead, Click on "1000 Samples" to the right of the second panel. The computer will take 1000 random samples (you won't see the individual values for each sample in the second panel) and plot the 1000 sample means in the third panel. This is the sampling distribution of the sample mean from samples of size 5. It should look somewhat, but not exactly, Normally distributed. The Central Limit Theorem is at work here, but the sample size is too small to make the sampling distribution appear completely Normal. The goal of this assignment is to explore the Central Limit Theorem and see how the sampling distribution of the sample mean is affected by the population and by the sample size.
(a) Select "Custom" from the pop-up menu at the top right of the window. Using the mouse, you can draw any population you wish in the top panel. Try drawing various population distributions, then taking 1000 random samples of size n = 5, and seeing if the sample means are Normally distributed. Hand in a printout of a window showing an example of a population for which the sample means are clearly NOT Normally distributed (using n = 5).
(b) Change the sample size to n = 25 using the pop-up menu at the right of the third panel. Simulate another 1000 samples from the same population distribution. Hand in a printout of this window as well. Are the sample means still non-Normally distributed? How does the sampling distribution of the sample mean change as you increase the sample size?