12.1 review
The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 219 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer. Complete parts (a) through (c) below
(a) Because these data are meant to prove that someone is guilty of fraud, what would be an appropriate level of significance when performing agoodness-of-fit test? Use α=*0.01.* (b) Using the level of significance chosen in part (a), test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey Benford's Law? What are the null and alternative hypotheses? *H0: The distribution of the first digits in the allegedly fraudulent checks obeys Benford's Law. H1:The distribution of the first digits in the allegedly fraudulent checks does not obey Benford's Law.* What is the test statistic? *χ20=28.571*(Round to three decimal) What is the P-value of the test? *P-value=0.000* (Round to three decimal places) Using the P-value approach, compare the P-value with the given α=0.01level of significance. Based on the results, do the first digits obey Benford's Law?*Reject the H0 because the calculated P-value is Les than the given α level of significance.* (c) Based on the results of part (b), could one think that the employee is guilty of embezzlement?*Yes, the first digits do not obey Benford's Law.* (write relevent theory/ statcrunch route/ formulas)
A researcher wanted to determine whether certain accidents were uniformly distributed over the days of the week. The data show the day of the week for n=301 randomly selected accidents. Is there reason to believe that the accident occurs with equal frequency with respect to the day of the week at the α=0.05 level of significance? Click the icon to view the table.
*Let pi = the proportion of accidents on day i, where i = 1 for Sunday, i = 2 for Monday, etc. What are the null and alternative hypotheses? *H0:p1=p2=...=p7=1/7 H1:At least one proportion is different from the others.* Compute the expected counts for day of the week (compute and draw table.) What is the test statistic? χ20=*15.35* (Round to three decimal) What is the P-value of the test? P-value=*0.018*(Round to three decimal places) Based on the results, do the accidents follow a uniform distribution? *Reject H0, because the calculated P-value is less than the given α level of significance. (write relevent theory/ statcrunch route/ formulas)
State the requirements to perform a goodness-of-fit test.
*all expected frequencies≥1* *at least 80% of expected frequencies ≥5*
According to the manufacturer of M&Ms, 13% of the plain M&Ms in a bag should be brown, 14% yellow, 13% red, 24% blue, 20% orange, and 16% green. A student randomly selected a bag of plain M&Ms. He counted the number of M&Ms of each color and obtained the results shown in the table. Test whether plain M&Ms follow the distribution stated by the manufacturer at the α=0.05 level of significance. (Table in question)
Determine the null and alternative hypotheses. Choose the correct answer below. *H0: The distribution of colors is the same as stated by the manufacturer. H1:The distribution of colors is not the same as stated by the manufacturer.* Compute the expected counts for each color. *Compute and draw table* (write relevent theory/ statcrunch route/ formulas)
Determine the expected count for each outcome. (Table in question)
The expected count for outcome 1 is 70.5. (Round to two decimal places as needed.) The expected count for outcome 2 is 183.3 (Round to two decimal places as needed.) The expected count for outcome 3 is 117.5 (Round to two decimal places as needed.) The expected count for outcome 4 is 98.7 (Round to two decimal places as needed.) (relevent theory,formulas statcrunch route)
Suppose there are n independent trials of an experiment with k>3 mutually exclusive outcomes, where pi represents the probability of observing the the probability of observing the outcome. What would be the formula of an expected count in this situation?
The expected counts for each possible outcome are given by Ei=np Subscript i. (write out formula)
Determine whether the statement below is true or false. The chi-square distribution is symmetric.
The statement is false. The chi-square distribution is skewed to the right.
Explain why chi-square goodness-of-fit tests are always right tailed.
The chi-square goodness-of-fit tests are always right tailed because the numerator in the test statistic is squared, making every test statistic, other than a perfect fit, positive.
If the expected count of a category is less than 1, what can be done to the categories so that a goodness-of-fit test can still be performed?
Two of the categories can be combined, or the sample size can be increased.
