AP Statistics - Test 3 - Answers

Workshop Statistics/Unit 3

Topics 11-15

BPS 3.1, 3.2

Mr. Coons

1. Circle the correct answer:

a. An observational study can show a causal relationship.

**b. An experimental study can show a causal relationship.**

c. The closer the value of r^2 is to 1, the more evidence there is of a causal relationship between the explanatory variable and the response variable.

d. Both a & b are true.

e. Both b & c are true.

2. Circle the correct answer. The design of an experiment is biased if:

a. A sample has large variability.

b. The center of a sample is not close to the population center.

c. All samples have large variability.

**d. The centers of all samples are on the same
side of the population center.**

e. Both c & d are true.

3. An educational researcher wants to compare the
effectiveness of using different computer set-ups to help in reading comprehension.
First she gives 12 students a reading comprehension test. Then she randomly
assigns them to computers with different set-ups. The computers have one
of two different size monitors (13 inch and 17 inch) and they display the
text at one of three different speeds (20 words per minute, 40 words per
minute & 80 words per minute). She conducts an experiment and then retests
the students and compares the increase in reading ability in each group.

a. What are the factors in this experiment?

screen size and speed

b. List the treatments in this experiment.

(13 in, 20 wpm), (13 in, 40 wpm), (13 in, 80 wpm), (17 in, 20 wpm), (17 in, 40 wpm), (17 in, 80 wpm)

c. Why is this study called an experiment?

Because specific treatments were imposed to observe the effect of the response variable.

d. The 12 students are listed below along with a set of random digits.

Anderson (01), Baxer(02), Cote(03), Fernandex(04),
Frank(05), Hicks(06), Klassen(07), Mihalko(08), Rustagi(09), Tomis(10),
Ulee(11), Zeg(12) |

33|06|3 4|18|42| 81|06|8 7|10|35| 09|00|1 0|33|67|
49|49|7 5|45|80| 81|50|7 2|71|02| 56|02|7 5|58|92. |

Demonstrate your understanding of simple random sampling by using the random digits to determine which of the 12 would be the first three randomly assigned. Briefly make it clear how your selections were made.

Number the students using two digit numbers from 01 to 12. Divide the random table into consecutive two digit blocks. Pick a random starting place in the table. I happened to pic the second group to start at. The first three 2 digit numbers between 01 and 12 without repetition are 06, 10, and 09. So the first three students would be Hicks, Tomis, and Rustagi

4. Identify and give a one sentence explanation of the three basic principles of Experimental Design.

see BPS

5. A scientist claims he has performed an experiment in which he both 1) uses a block design, and 2) uses an SRS of ENTIRE population. Explain why this is not possible. Illustrating your point with an example is acceptable.

Craig Lund suggests the following example: "Suppose you were doing an experiment on 30 rats in which you gave them 1 of 3 possible diets and then recorded their weights after 5 weeks. In a block design you would divide the 30 rats into 10 groups of 3 rats of similar weights and then randomly allocate which diet they would have." However, using this block method there is no chance that each diet has say, and equal distribution of rates of different weights. In other words, the probability of every possible sample is not the same as required by the definition of SRS.

6. Bill, a statistician, said that the temperature was so cold yesterday at the North pole that it was 3.5 standard deviations BELOW normal. He said that this was a statistically significant event. Clearly demonstrating your understanding of the terms "statistically significant" and including numeric support to explain if he was correct.

Brian O'Connor suggests that "Bill was correct in saying the temperature was statistically significant because it is included in the definition as being "unlikely to occur by chance alone." The likelihood [of] getting a temperature 3.5 standard deviations or more below normal is normalcdf(-1000000,-3.5,0,1) = 0.000233 or about 0.023%, which is not likely to occur just by chance [very often].

7. A gambler has a special coin that has been flipped so many times that he knows over the long, long run it lands heads 55 out of 100 times.

a. Fully annotating, working by hand (i.e., without using the statistical functions of your calculator), and using the supplied table determine the probability of a sample of 20 flips having 6 or fewer heads.

b. Draw a diagram, write down some probability notation and then, using any method you wish, determine the probability of a sample of 20 flips having between 8 and 14 heads.

c. How many flips from a sample of 20 flips would be expected to be heads if the probability of getting that many heads was at least 10% more than the probability of getting the population parameter? Show your work.

0.578*20 flips = 11.56 flips. To have AT LEAST 10% more then we need 12 flips |

8. Part of the conclusion of the *CLT for Sample
Proportions *states that "the sampling distribution of the sample
proportion p-hat is approximately normal". ANSWER ONLY ONE OF THE FOLLOWING
TWO CHOICES:

- Explain what a "sampling distribution of the sample proportion p-hat " is.

The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

- Explain why we need to know the shape of the sampling distribution of the sample proportion p-hat

From Jaime Bard: "We are only accustomed to normal curves, this is, symmetric, bell shaped single-peaked curves. Not all data fits within this "normal" model. Therefore, it is important to know the shape of the distribution to use the best model for inferential use. It would not be worth while to use a normal curve [to describe a population that had a sampling distribution] which looked like a sine curve since it would be quite inaccurate.

9. Considering 1) the definition of *confidence*
and 2) the *CLT for Proportions* and 3) assuming no other changes in
values, explain the effect of increased sample size on *confidence*.
Make sure your explanation includes what you have confidence in.

From Mike: Increased sample size reduces the variability of the sampling distribution of p-hat. The standard deviation is decreased because according to the CLT, std dev = r(f(q(1-q),n)) so as n increases the standard deviation decreases. Because of this confidence (that a sample proportion is within a certain distance of the actual population proportion) is also increased. That is, the distances that the sample proportion is within have decreased so you are more confident that the sample proportion is within a smaller distance of the actual population proportion.

Extra Credit - From Chris Wells (3 points - all or nothing)

The Human Resources Department at a company with
10,000 employees suspects that over the last 365 days employees have been
taking sick days more often on Mondays and Fridays so that they have three
consecutive days off. The HR department takes a SRS of 500 sick days and
finds that 40.2% of those were taken on Mondays or Fridays. What does the
Human Resources Department conclude and why?