1. (9) A tetrahedron is a fully-symmetric pyramid-like solid, with four faces made up of equal-sized equilateral triangles. Suppose that the four faces contain different number of spots, from one spot to four spots, so that the tetrahedron may be rolled as a die. The number of spots which results from the roll of this die is the number of spots on the bottom face, the one which can not be seen. In the example, the die has thrown a two.
a. Suppose two such dice are rolled. Write a list of all possible equally-likely pairs of numbers of spots which could result.
b. If two such dice are rolled, what is the probability that the sum of the spots will be four?
c. If two such dice are rolled, what is the probability that the sum of the spots will be greater than two?
2. (10) Prior to an election, campaign committees or news organizations sometimes pay for an opinion poll of prospective voters. Suppose that a newspaper reports that, if the election were held now, 52% of voters would vote for candidate X, with a margin of error of plus or minus 3%.
a. If no confidence level is reported for the result, what is the assumed level?
b. What is the minimum number of randomly-selected prospective votes who needed to be polled in order to achieve the level of confidence you assumed in part (a)?
c. Suppose that the poll was conducted as you specified in part (b). Explain what difference it would make to your confidence in the statement "52% + 3% of voters would choose candidate X," if the poll were used to predict a local election, versus a national one.
d. Discuss whether the poll result implies that candidate X will win a majority of the votes.
3. (7) In Data Set 11 of Appendix C of the textbook are recorded the inches of rain which fell on the different days of the week for the fifty-two weeks of one year in Boston. Use this data and a 10% significance level to test the idea that the average rainfall is not the same on all the days of the week in Boston. In justifying your conclusion, include an explanation in plain English.
4. (19) A quality control inspector randomly removed a small sample of light bulbs from an assembly line and turned them on, to see how long they would burn. Here are the numbers of hours her experiment produced: 972, 854, 952, 893, 945, 909, 850, 918, 909, 980, 926, 883, 925.
a. The first thing the inspector wants to know is whether she can be 95% certain that the average bulb from the factory burns at least 900 hours, as stated on its package.
i. What is the null hypothesis for her hypothesis test?
ii. What is the significance level of the test?
iii. What is the value of the test statistic she computes?
iv. What value of the test statistic is critical?
v. What numbers could she compare in order to make up her mind?
vi. How could she express her conclusion in language which any literate person could understand?
b. The second thing the inspector is worried about is if there is too much variation among the lifetimes of bulbs. She believes that the factory standard deviation should be no more than 35 hours.
i. What sort of statistical test should settle this question?
ii. Perform the test you selected, and explain to the inspector your basis for saying whether or not she has cause to be worried about the amount of variation.
5. (8) Gregor Mendel crossed two breeds of peas and examined the crop. He found 892 wrinkled green peas, 378 smooth green peas, 618 wrinkled yellow peas, and 241 smooth yellow peas.
a. If he picked a smooth pea at random from his crop, what is the probability that it was yellow?
b. If he picked a pea at random from his crop, what is the probability that it was yellow or smooth?
c. If he picked a pea at random from his crop, what is the probability that it was green and wrinkled?
6. (25) Data set 28 in Appendix C of the textbook shows the weight, in grams, of a sample of sugar packets.
a. What is the sample standard deviation?
b. What is the median of the sample?
c. Give a boxplot of the sample.
d. Give a histogram with seven classes for the sample.
e. Test whether the sample came from a normally-distributed population, and justify your conclusion.
f. Give a 92%-confidence interval for the mean of the population of packets from which this sample was drawn.
7. (8) a. If you know that 45% of the human population has black hair, and if you happened across a group of 25 people of whom 17 had black hair, would you consider this group unusual?
b. By what standard do you justify your answer to part (a)?
8. (6) Here is a probability distribution for a random variable X, with the last line incomplete.
a. Complete the last line of the table.
b. Find the mean of the probability distribution.
c. Find the standard deviation of the probability distribution.
9. (8) Data set 25 in Appendix C of the textbook shows miscellaneous annual statistics.
a. Write the linear regression equation predicting the number of US murders and non-negligent homicides as a function of the annual number of sunspots.
b. Make the best prediction of the number of murders and non-negligent homicides if the number of annual sunspots will be 40.
The solution provides step by step method for the calculation of descriptive statistics and probability . Formula for the calculation and Interpretations of the results are also included.