Hypothesis testing homework solutions

Setting up and interpreting results of a hypothesis test
ACT-1. Practice with p-values
Use the density tool to practice finding p-values.
I. Set the mean to 35, the standard deviation to 12.5, and the sample size to 100. What is the probability of finding a value:
a. less than 33 .055
b. greater than 37.1 .039
c. less than 32.8 or greater than 36.9 .039+.065=.104

II. Change the sample size to 250. Now that the sample size is greater, find the following probabilities and compare them with those found in the first question:
a. less than 33 .006
b. greater than 37.1 .004
c. less than 32.8 or greater than 36.9 .003+.008=.011
Larger samples taken from the same population have a much smaller chance to have a mean value far from the population mean. Increasing the sample size by a factor of 2.5 decreased the likelihood of these rare events by a factor of 10.
III. Keeping the sample size at 250 and the mean at 35, try changing the standard deviation to 11 (a difference of only 1.5 from the previous standard deviation) and compare these results with the results you obtained in the second question.
a. less than 33 .034
b. greater than 37.1 .029
c. less than 32.8 or greater than 36.9 .023+.042=.065
A relatively small, about 10% decrease in the standard deviation had a huge effect increasing (by a factor of 6) the chances for certain sample mean values to come up that are far form the population mean..
IV. Set the mean to 34, the standard deviation to 12.5, and the sample size to 100. Find the following probabilities and compare them with the ones you found in the first question.
a. less than 33 .21
b. greater than 37.1 .007
c. less than 32.8 or greater than 36.9 .167+.01=.177
33 is much closer (almost 1.25 closer which would be a full standard deviation closer) to the mean now so chances for a sample mean to be 33 increased dramatically. On the other and 37.1is a full standard deviation further than where it was when the mean was 35. therefore fewer samples show this value. In part c) the lower value got closer the higher value mover further form the mean so it evens out to about the same chance overall as in the corresponding part in I.
Write a brief summary of what you observed when certain values were changed.

MRA-1. Population Mean Hypotheses
Each of the following paragraphs calls for a statistical test about a population mean m. State the null hypothesis H0 and the alternative hypothesis Ha in each case.

(a)The diameter of a spindle in a small motor is supposed to be 5 mm. If the spindle is either too small or too large, the motor will not work properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target.
H0: ΅=5
Ha: ΅≠5

(b)Census Bureau data show that the mean household income in the area served by a shopping mall is \$42,500 per year. A market research firm questions shoppers at the mall. The researchers suspect the mean household income of mall shoppers is higher than that of the general population.

H0: ΅=42,500\$
Ha: ΅>42,500\$

©The examinations in a large accounting class are scaled after grading so that the mean score is 50. The professor thinks that one teaching assistant is a poor teacher and suspects that his students have a lower mean score than the class as a whole. The TA's students this semester can be considered a sample from the population of all students in the course, so the professor compares their mean score with 50.
H0: ΅=50
Ha: ΅<50

MRA-4. Explaining Ethnocentrism
A social psychologist reports that in our sample, "ethnocentrism was significantly higher (p= 0.05) among church attenders than among nonattenders."
Explain what this means in language understandable to someone who knows no statistics. Do not use the word "significance" in your answer.

Researchers found that the percentage of ethnocentrics was higher among church attenders compared with the same percentage for non-attenders. In fact the evidence for that was so extreme that if there was no difference between the percentage of ethnocentrics between attenders and non-attenders we would see such an extreme result only 5% of