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Aw, nuts! A company claims that each batch of its deluxe mixed nuts contains $52\%$cashews, $27\%$almonds,$13\%$macadamia nuts, and 8% brazil nuts. To test this claim, a quality control inspector takes a random sample of $150$nuts from the latest batch. The one-way table below displays the sample data.

(a) State appropriate hypotheses for performing a test of the company鈥檚 claim.

b) Calculate the expected counts for each type of nut. Show your work.

Mars, Inc., reports that their M&M鈥橲 Peanut Chocolate Candies are produced according to the following color distribution: $23\%$each of blue and orange, $15\%$each of green and yellow, and $12\%$each of red and brown. Joey bought a bag of Peanut Chocolate Candies and counted the colors of the candies in his sample: $12$blue, $7$orange,$13$green, $4$yellow, $8$red, and $2$brown.

State appropriate hypotheses for testing the company鈥檚 claim about the color distribution of peanut $M\&M鈥�S$.

Representative sample? For a class project, a group of statistics students is required to take an SRS of students from their large high school to take part in a survey. The students鈥� sample consists of $54$freshmen, $66$sophomores, $56$juniors, and $30$seniors. The school roster shows that $29\%$of the students enrolled at the school are freshmen, $27\%$are sophomores, $25\%$ are juniors, and $19\%$ are seniors.

(a) Construct a well-labeled bar graph that shows the distribution of grade levels (in percents) for the sample data. Do these data give you any reason to suspect that the statistics students鈥� sample is unusual? Explain. (b) Use an appropriate test to determine whether the sample data differ significantly from the actual distribution of students by grade level at the school.

A chi-square goodness-of-fit test is used to test whether a 0 to 9 spinner is "fair" (that is, the outcomes are all equally likely). The spinner is spun 100 times, and the results are recorded. The degrees of freedom for the test will be

(a) 8 .

(c) 10 .

(e) None of these.

(b) 9 .

(d) $99.$

A company claims that each batch of its deluxe mixed nuts contains $52\%$cashews, $27\%$almonds, $13\%$macadamia nuts, and $8\%$brazil nuts. To test this claim, a quality control inspector takes a random sample of $150$nuts from the latest batch. The one-way table below displays the sample data.

(a) State appropriate hypotheses for performing a test of the company鈥檚 claim.

(b) Calculate the expected counts for each type of nut. Show your work

No chi-squareThe principal in Exercise $9$also asked the random sample of students to record whether they did all of the homework that was assigned on each of the 铿乿e school days that week. Here are the data:

Explain carefully why it would not be appropriate to perform a chi-square goodness-of-铿乼 test using these data.

A Type I error would occur if we conclude that

(a) HCC wiring caused cancer when it actually didn't.

(b) HCC wiring didn't cause cancer when it actually did.

(c) there is no association between the type of wiring and the form of cancer when there actually is an association.

(d) there is an association between the type of wiring and the form of cancer when there actually is no association.

(e) the type of wiring and the form of cancer have a positive correlation when they actually don't.

Benford鈥檚 lawFaked numbers in tax returns, invoices, or expense account claims often display patterns that aren鈥檛 present in legitimate records. Some patterns are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the 铿乺st digits of numbers in legitimate records often follow a model known as Benford鈥檚 law.^$3$ Call the 铿乺st digit of a randomly chosen record X for short. Benford鈥檚 law gives this probability model for X (note that a 铿乺st digit can鈥檛 be 0):

A forensic accountant who is familiar with Benford鈥檚 law inspects a random sample of invoices from a company that is accused of committing fraud. The table below displays the sample data.

(a) Are these data inconsistent with Benford鈥檚 law? Carry out an appropriate test at the $伪=0.05$level to support your answer. If you 铿乶d a signi铿乧ant result, perform follow-up analysis.

(b) Describe a Type I error and a Type II error in this setting, and give a possible consequence of each. Which do you think is more serious?

A large distributor of gasoline claims that $60\%$all cars stopping at their service stations choose regular unleaded gas and that premium and supreme are each selected $20\%$of the time. To investigate this claim, researchers collected data from a random sample of drivers who put gas in their vehicles at the distributor's service stations in a large city. The results were as follows:

Carry out a significance test of the distributor's claim. Use a $5\%$significance level.

Mars, Inc., reports that their M&M鈥橲 Peanut Chocolate Candies are produced according to the following color distribution: 23% each of blue and orange, 15% each of green and yellow, and 12% each of red and brown. Joey bought a bag of Peanut Chocolate Candies and counted the colors of the candies in his sample: 12 blue, 7 orange, 13 green, 4 yellow, 8 red, and 2 brown

State appropriate hypotheses for testing the company鈥檚 claim about the color distribution of peanut M&M鈥橲