Question 1:
1) Suppose there is an income tax scheme under which a worker receives 80% of the original pre-tax earnings and then is given a (tax-free) grant of $2000.
a) Write the linear equation that describes this relationship between after-tax earnings (call it random variable Y) and pre-tax earnings (call it random variable X):
b) Given a normal distribution where X ∼ N ($50,000, $9,000), what is the expected value of the individual's after-tax earnings?
c) What is the variance of the distribution in part b?
Question 2:
2) Let X indicate the number of "tails" that occur when two coins are tossed.
a) The probability distribution of a discrete random variable is a list of all possible values of the variables and the pr ob ability that each value will occur. Derive the probability distribution of X in the chart:
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Outcome (number of tails)
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X = 0
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X = 1
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X = 2
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Probability
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b) The emulative probability distribution is the probability that the random variable is less than or equal to a particular value. The bottom row in the dart below gives the cumulative probability distribution of the random variable X. Derive the cumulative probability distribution of X in the chart:
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Outcome (number of tails)
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X < 0
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0 ≤ X < 1
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1 ≤ X < 2
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X ≥ 2
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Probability
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c) Derive the mean of X (the average number of tails we expect). Show your work:
d) Derive the variance of X. Slow your work:
Question 3:
3) Michael Sc ott1 the manager of Dunder-Mifflin Inc, a paper distribution company, claims that his salesmen make an average of 40 sales calls per week on potential clients. The sales representatives Dwight Schrute and Jim Halpert say that this estimate is too low. To investigate, a random sample of 28 sales representatives from the regional branches reveals that the mean number of calls made last week was 42. The standard deviation of the sample is 2.1 calls. Using the .05 significance level, can we conclude that the mean number of calls per salesperson per week is more than 40?
1) Write the null and alternate hypothesis:
2) Calculate the test statistic:
3) Draw the critical region. Be sure to mark both the critical value (s) and the test statistic. When will we reject the null hypothesis (in other words, what is the decision rule)?
4) What do you conclude? Why? (Test statistic compared to critical value (s)).
5) Find the p-value, Be sure to draw what this looks like.
Question 4:
4) An automated assembly line is set to fill a small bottle with 9.0 grams of cough syrup. A sample of eight bottles revealed the following amounts (grams) in each bottle:
9.2 8.7 8,9 8.8 8.5 8.7 9.0 8.6
At the .01 significance level can we conclude that the mean weight is less than 9.0 grams?
a) State the null and the alternate hypothesis:
b) How many degrees of freedom are there? Show your work.
c) Give the decision rule:
d) Compute the value of t.
e) What is your decision regarding the null hypothesis? Why?
f) Estimate the p-value.