Chi-Square Goodness-of-Fit Tests f 7) Birds in the trees Researchers studied the behav or of birds that were searching for seeds and insects i in an Oregon forest. In this forest, 54% of the trees were Douglas firs, 40% were ponderosa pines, and 6% were other types of trees. At a randomly selected time during the day, the researchers observed 156 red-breasted nuthatchesr70 were seen in Douglas firs, 79 in ponderosa pines, and 7 in other types of trees. 2 Do these data suggest that nuthatches prefer particular types of trees when they're searching for seeds and insects? Carry out a chi-square goodness-of-fit test to help answer this question. ^ Ct m & <?. Benford's law Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren't 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 first digits of numbers in legitimate records often follow a modttHrrtowTT as &enfordVtaw-.3-CaU the-first digit of a randomly chosen record X for short. Benford's law gives this probability model for X (note that a first digit can't be 0): (?) No chi-square A school's principal wants to know , if students spend about the same amount of time ; on homework each night of the week. She asks a random sample of 50 students to keep track of their homework time for a week. The following table displays tha^average arnount_pf time (in minutes)^ students reported per night: Night: Sunday Monday Tuesday Wednesday Thursday Friday Saturday Average 130 108 115 104 99 37 62 time: Explain carefully why it would not be appropriate to perform a chi-square goodness-of-fit test using these data. Jc4 a 4.0 ere. pg688 Mendel and the peas Gregor Mendel (1822-1884), an Austrian monk, is considered the father of genetics. Mendel studied the inheritance of various traits in pea plants. One such trait is whether the pea is smooth or wrinkled. Mendel predicted a ratio of 3 smooth peas for every 1 wrinkled pea. In one experiment, he observed 423 smooth and 133 wrinkled peas. The data were produced in such a way~"thatlKe~Randorn and Independent conditions are met. Carry out a chi-square goodness-of-fit test based on Mendel's prediction. What do you conclude? VA i 0 I First digit (X): 1 2 3 4 5 6 7 8 ° ' Probability: 0.301 0.176 0.125 0.097 0.079 0.067 0058 0051 0046 A forensic accountant who is familiar with Benford's law inspects a random sample of250 invoices from a company that isaccused oFcommitting fraud. The table below displays the sample data. 1 ^ | First digit: i Count: I 1 61 2 50 3 43 4 34 5 25 6 16 7 [ 1 i } 9 5 6 - (a) Are these data inconsistent with Benford's law? Carry out an appropriate test at the a = 0.05 level to support your answer. If you find a significant result, perform a follow-up analysis. (b) Describe a Type 1 error and a Type II error in this setting, and give a possible consequence of each. Which do you think is more serious? 11)
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Chi-Square Goodness-of-Fit Tests
f 7) Birds in the trees Researchers studied the behav orof birds that were searching for seeds and insects iin an Oregon forest. In this forest, 54% of the treeswere Douglas firs, 40% were ponderosa pines,and 6% were other types of trees. At a randomlyselected time during the day, the researchersobserved 156 red-breasted nuthatchesr70 wereseen in Douglas firs, 79 in ponderosa pines, and7 in other types of trees.2 Do these data suggest thatnuthatches prefer particular types of trees whenthey're searching for seeds and insects? Carry outa chi-square goodness-of-fit test to help answer thisquestion.
^Ct
m&<?.
Benford's law Faked numbers in tax returns, invoices,or expense account claims often display patterns thataren't present in legitimate records. Some patterns areobvious and easily avoided by a clever crook. Othersare more subtle. It is a striking fact that the firstdigits of numbers in legitimate records often followa modttHrrtowTT as &enfordVtaw-.3-CaU the-first digitof a randomly chosen record X for short. Benford'slaw gives this probability model for X (note that a firstdigit can't be 0):
(?) No chi-square A school's principal wants to know ,if students spend about the same amount of time ;on homework each night of the week. She asks arandom sample of 50 students to keep track of theirhomework time for a week. The following table
displays tha^average arnount_pf time (in minutes)^students reported per night:
Night: Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Average 130 108 115 104 99 37 62time:
Explain carefully why it would not be appropriate toperform a chi-square goodness-of-fit test using thesedata.
Jc4 a
4.0 ere.
pg688
Mendel and the peas Gregor Mendel (1822-1884),an Austrian monk, is considered the father ofgenetics. Mendel studied the inheritance of varioustraits in pea plants. One such trait is whether thepea is smooth or wrinkled. Mendel predicted aratio of 3 smooth peas for every 1 wrinkled pea. Inone experiment, he observed 423 smooth and 133wrinkled peas. The data were produced in such away~"thatlKe~Randorn and Independent conditionsare met. Carry out a chi-square goodness-of-fittest based on Mendel's prediction. What do youconclude?
VA i0 I First digit (X): 1 2 3 4 5 6 7 8
° ' Probability: 0.301 0.176 0 .125 0.097 0.079 0.067 0 0 5 8 0 0 5 1 0046
A forensic accountant who is familiar with Benford'slaw inspects a random sample of250 invoices froma company that isaccused oFcommitting fraud. Thetable below displays the sample data.
1̂| First digit:
i Count:I
1
61
2
50
3
43
4
34
5
25
6
16
7 [
1 i
} 9
5 6 -
(a) Are these data inconsistent with Benford's law?Carry out an appropriate test at the a = 0.05 level tosupport your answer. If you find a significant result ,perform a follow-up analysis.
(b) Describe a Type 1 error and a Type II error inthis setting, and give a possible consequence of each.Which do you think is more serious?
11)
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