Download Activity 9A The height of young women varies approximately

Activity 9A
The height of young women varies approximately according to the N (64.5, 2.5) distribution. That is to
say, the population of young women is normally distributed with mean µ = 64.5 inches and standard
deviation σ = 2.5 inches. The random variable measured is X = the height of a randomly selected young
woman. In this activity you will use the TI-84 to sample from this distribution.
1. If we choose one woman at random, the heights we get in repeated choices follow the N (64.5, 2.5)
distribution. On your calculator, go into the statistics/List Editor and clear L1. Simulate the heights of
100 randomly selected young women and store these heights in L1 as follows:
Place your cursor at the top of L1 (highlight L1)
Press [MATH], choose PRB, and choose 6: randNorm (.
Complete the command: randNorm (64.5, 2.5, 100) and press [ENTER].
2. Plot a histogram of the 100 heights as follows. Deselect active functions in the Y = window, and turn
off all STAT PLOTS. Set WINDOW dimensions to: X[57, 72]2.5 and Y[-10, 45]5 to extend three standard
deviations to either side of the mean, 64.5. Define PLOT 1 to be a histogram using the heights in L1.
Press [GRAPH] to plot the histogram. Describe the approximate shape of your histogram. Is it fairly
symmetric or clearly skewed?
3. Approximately how many heights should there be within 3σ of the mean (i.e., between 57 and 72)?
Use TRACE to count the number of heights within 3σ. How many heights should there be within 1σ of
the mean? Within 2σ of the mean? Again use TRACE to find these counts, and compare them with the
numbers you would expect.
4. Use 1-Var Stats to find the mean, median, and standard deviation for your data. Compare ̅ with the
population mean µ = 64.5. Compare the sample standard deviation s with σ = 2.5. How do the mean
and median for your heights compare? Recall that the closer the mean and the median are, the more
symmetric the distribution.
5. Define PLOT 2 to be a boxplot using L1 and then GRAPH again. The boxplot will be plotted above the
histogram. Does the boxplot appear symmetric? How close is the median in the boxplot to the mean of
the histogram? Based on the appearance of the histogram and the boxplot, and a comparison of the
mean and median, would you say that the distribution is nonsymmetric, moderately symmetric, or very
symmetric?
6. Repeat steps 1 to 5 three times. Each time, record the mean ̅ , median, and standard deviation s.
(Note: While this is going on, Mr. Song will draw a baseline at the bottom of a whiteboard and mark a
scale from 63 to 66 with tick marks at 0.25 intervals. The tick marks should be spaced about an inch
wider apart than the width of the Post-it Notes. Each tick mark will represent the center of a bar in a
histogram.)
7. Write (big and neat) the mean ̅ for each sample on a different Post-it Note. Next, you will build a
“Post-it Note histogram” of the distribution of the sample means ̅ . When instructed, go to the board
and stick each of your notes directly above the tick mark that is closest to the mean written on the note.
When the Post-it Note histogram is complete, answer the following questions.
(a) What is the approximate shape of the distribution of ̅ ?
(b) Where is the center of the distribution of ̅ ? How does this center compare with the mean
of heights of the population of all young women?
(c) Roughly, how does the spread of the distribution of ̅ compare with the spread of the
original distribution (σ = 2.5)
8. While someone calls out the values of ̅ from the Post-it Notes, enter these values in L2 in your
calculator. Turn off PLOT 1 and define PLOT 3 to be a boxplot of the ̅ data. How do these distributions
of X and ̅ compare visually? Use 1-var stats to calculate the standard deviation sx for the distribution of
̅ Compare this value with
√
.
9. Fill in the blanks in the following statement with a function of µ or σ: “The distribution of ̅ is
approximately normal with mean µ( ̅ =_________ and standard deviation σ( ̅ =_____________”