Midterm Practice Solutions
Stat 200 Corwin
1. For each part, answer with the population, a sample, data, a statistic, or a parameter.
(a) To estimate the number of New Yorkers who smoke cigarettes, John surveyed 100 peo-ple. The 100 people are: a sample
(b) Jane determined the average salary of the five janitors who work for Ed’s Cleaning Service. The five janitors are: the population
(c) Jane determined the salaries of the five janitors who work for Ed’s Cleaning Service. They are (in dollars)
11,500 11,900 12,200 12,350 12,900 These five numbers are: data
(d) For a month, Jesse identified every insect he could catch in his apartment. He found that 18% were spiders. The 18% is: a statistic (no way he got every bug)
(e) Jane determined the marital status of the five janitors who work for Ed’s Cleaning Ser-vice. She found that three were married, one was divorced, and one was widowed. These results are: data
2. Indicate whether the data described is discrete or continuous. (a) Size of a painting, in square inches: continuous
(b) Capacity of a car’s gas tank: continuous
(c) Number of angels that can sit on the head of a pin: discrete (d) Number of raindrops that fall on Virginia in one year: discrete
(e) Amount of rain that falls on Virginia in one year, in cubic centimeters: continuous
3. Which of the following are measures of central tendency and which are measures of dis-persion?
mean: central tendency median: central tendency range: dispersion variance: dispersion std dev: dispersion
Remember: mean, median, and mode are our only measures of central tendency, and range, variance, standard deviation, and IQR are our only measures of dispersion/spread/variation.
4. 15.27 21.52 23.64 0.19 22.10 16.29 For the data above:
(a) Find the mean. 16.5 (approx) (b) Find the median.
18.9
(c) Construct the five-number summary.
(min, Q1, Q2, Q3, max) = (0.19, 15.27, 18.9, 22.10, 23.64)(0.19, 15.27, 18.9, 22.10, 23.64)(0.19, 15.27, 18.9, 22.10, 23.64) (d) Construct a box-and-whisker plot.
(e) Does 0.19 seem to be an outlier? Looks like it to me.
(f) If 0.19 is an outlier, which measure of central tendency should be used to represent the data?
the median
(g) Assume that the data shows the amount of rain, in mm, for six different Sundays in 2009, and that you are interested in the rainfall for that entire year. Compute the standard deviation.
This means that the data is a sample, so std dev = s = q
Σ(x− ¯x)2
n−1 ≈ 8.668.668.66. (Do this on your calculator.)
5. Jim has observed the models of Canon cameras carried by 42 tourists and compiled the following table:
Model Frequency Powershot 320 HS 27 Powershot S410 12 Rebel 3 1 Rebel 6 1 Rebel T7i 1
6. Janet has found that the mean of her data is 101.3 and that the standard deviation is 14.7. (a) What is the z-score of the data value x = 117?
z=x− µ
σ =
117 − 101.3
14.7 ≈ 1.071.071.07 (b) What data point corresponds to the z-score −1.45?
−1.45 =x− 101.3
14.7 ⇒ x − 101.3 = 14.7(−1.45) ⇒ x = 79.98579.98579.985
7. The Grand Island, NE school district administered the Iowa Test of Basic Skills to 4696 students in 2007. Janelle’s daughter’s score put her in the 63rdpercentile.
(a) Does this mean that the girl got about 63% of the questions right? No.
(b) What does it mean?
About 63% of the students scored lower than Janelle’s daughter. 8. Find the mean of the following frequency distribution.
Remember that this table simply means that the data consists of seven 4s, four 5s, and so on. Below, I’ve extended the table to show (data value × frequency). To get the mean, simply add those values up and divide by the total number of data points.
Data value Frequency value × frequency
4 7 28 5 4 20 6 6 36 7 3 21 8 5 40 Σ f = 25 sum = 145 Mean = 14525 = 5.85.85.8 (b) Find the median.
25 points ⇒ mid pos is 25+12 = 13 13thdata point is a 6 ⇒ median = 6 (c) Find the mode.
9. Decide whether each of the distributions shown is closest to a uniform, symmetric, bimodal, skewed left, or skewed right distribution.
bimodal uniform
symmetric skewed left
10. Jerry measured the actual amount of soda in 241 twelve-ounce bottles and constructed the following relative frequency histogram:
Construct the frequency histogram for Jerry’s data. (You will have to make estimates.) I did this by estimating the number of bottles in each class. For example, it looks like about 2% of the bottles are in the first class, and 2% of 241 is 4.82, so I figured five bottles were in the first class. I’ve named the classes A, B, etc. to fit them on the graph.
11. The table below shows the number of sheep with each marking color that Siobhan observed on a hill yesterday.
Color Number of Sheep
Pink 56
Blue 47
No marking 18
What is the mode of this data? Pink
12. A TI calculator was used to compute summary statistics for a particular data set, with the results shown below.
(a) What is the range of the data? 1.362 − 0.851 = 0.5110.5110.511
(b) What is the interquartile range of the data? Q3− Q1= 0.1710.1710.171
13. Last semester, Jackson got a C in English, a B in Snorkeling, a B in Psychology, a C in Biology, and a D in French. Snorkeling was a one-credit class, and Biology was a four-credit class; all the others were three-credit classes. To the nearest hundredth, what was his GPA?
2 · 3 + 3 · 1 + 3 · 3 + 2 · 4 + 1 · 3
3 + 1 + 3 + 4 + 3 ≈ 2.072.072.07 14. Complete the following table:
Interval Frequency Relative Frequency Cumulative Frequency 0–2 10 103010301030 ≈ 0.333≈ 0.333≈ 0.333 10 2–4 0 0 10 4–6 4 0.133 14 6–8 9 0.3 23 8–10 7 0.233 30 ∑ f = 30
15. Jan asked a number of people to estimate the average number of steps they took each day. The table below summarizes the responses:
Number of Steps Frequency
0 – 2000 36 2000 – 4000 32 4000 – 6000 32 6000 – 8000 18 8000 – 10 000 8 10 000 – 12 000 0 12 000 – 14 000 0 14 000 – 16 000 1
If a histogram were made from this table:
(a) What shape would it have? skewed right (see histogram below) (b) Would it show a possible outlier? yes
16. What is the mode of the data set represented by the histogram below?
Orange and Blue (bimodal)
17. The standard deviation of a certain data set is 176.89. To the nearest hundredth, what is the variance of the data?
18. What value of c makes the table below a probability distribution?
xxx 1 2 3 4 5
P(X = x)P(X = x)P(X = x) 0.15 c 0.1 0.04 0.13
0.58
19. A bag contains several lipsticks which are indistinguishable except for color. There are three lipsticks of the “Ballerina Shoes” color, two of the “Blushing Berry” color, one “Brazil Nut,” and three “I Pink You’re Cute.” (Yes, these are real lipstick colors.) If a lipstick is drawn at random from the bag, what is the probability that it is named after a fruit or nut?
Three of the nine lipsticks are named after a fruit or nut, so the probability is 39 =131313. 20. Joan asked a sample of students at her college to identify their favorite pizza toppings and
found the following:
Favorite Topping Frequency
None 27
Pepperoni 25
Mushrooms 8
Anchovies 3
∑ f = 63
If one of the people interviewed is selected at random, what is the probability that the person’s favorite topping is pepperoni?
25 63 25 63 25 63 ≈ 0.3968
21. A single card is drawn from a standard deck, and a single die is rolled once. Let E be the event that the card is an ace, F the event that the card is a heart, and G the event that the die shows a number less than 3.
(a) P(E) =525252444 = 131 (There are four aces in a deck.) (b) P(F) =135213521352 =14 (There are 13 hearts in a deck.)
(c) P(G) =262626= 13 (“number less than 3” means a 1 or a 2) (d) What is the probability that both E and G occur?
22. Two dice are rolled, one red and one green. Define E= get a 2 on the red
F= get an even number on the red G= get a sum of 5
H= get a sum of 2
List all the pairs of mutually exclusive events. E, HE, HE, H F, HF, HF, H G, HG, HG, H
23. In how many ways can the 40 members of a 4H club select a president, a vice president, a secretary, and a treasurer, assuming that no one can hold more than one office at a time?
There are 40 ways to select a president. As the person selected to be president cannot hold any of the other offices, there are then 39 ways to select a vice president, and so on. Altogether there are 40 · 39 · 38 · 37 = 2, 193, 3602, 193, 3602, 193, 360 ways to choose the four officers. 24. In how many different ways can a person choose three different movies to watch in a theater
playing 11 different movies?
Order doesn’t matter here, so there are11C3= 165165165 ways.
25. An urn contains 12 balls identical in every respect except color. There are 3 red balls, 7 green balls, and 2 blue balls. You draw two balls from the urn but replace the first ball before drawing the second. Find the probability that the second ball is green if the first ball is red.
Because you replace the first ball, there are still 3 red balls, 7 green balls, and 2 blue balls in the urn when you go to draw out the second ball. Thus, the probability that the second ball is green is 121212777.
26. An urn contains 12 balls identical in every respect except color. There are 3 red balls, 7 green balls, and 2 blue balls. You draw two balls from the urn but do not replace the first before drawing the second. Find the probability that the second ball is green if the first ball is red.
After you take out the first ball, there are 2 red balls, 7 green balls, and 2 blue balls in the urn. Thus, the probability that the second ball is green is111111777.
27. Janice surveyed randomly chosen students about whether they preferred cats or dogs as pets and found the following:
Cats Dogs Total
Men 37 33 70
Women 34 36 70
Total 71 69 140
First, what is the sample space of this experiment? {Men/Cats, Men/Dogs, Women/Cats, Women/Dogs} If one of the people interviewed is selected at random: (b) What is the probability that the person prefers cats?
71 140 71 140 71 140
(c) If the person is female, what is the probability that she prefers cats?
Of the 70 females surveyed, 34 preferred cats, so the probability wanted is 347034703470. (d) What is the probability that the person is male if it is known that he or she prefers cats?
