A fast-food restaurant finds that its daily profits have a normal distribution with mean
US$140 and standard deviation US$80.
(1) Find the probability that the restaurant loses money on a given day. (6%)
(2) Find the probability that the restaurant makes money for the next seven days in a
row. (5%)
(3) What is the expected number of days that the restaurant makes money for the next
seven days in a row> (5%)
(4) What assumptions must you make for this calculation to be valid for parts (2) and
(3)? (4%)
For a large supermarket chain, a women’s group claimed that female employees were
passed over for management training in favor of their male colleagues. The company
denied this claim, saying that they picked the employees from the eligible pool at
random to receive this training. The large pool more than 1000 eligible employees who
can be tapped for management training is 40% female and 60% male. Since this
program began, 28 of the 40 employees chosen for management training were male and
12 were female. (30%)
(1) In a significant test, the random sampling assumption is the claim of the company.
Let p be the probability of selecting a male for any given selection. State the null
and alternative hypotheses for a test based on p to investigate the strength of
evidence to support the women’s claim. (3%)
(2) Conduct a statistical test for part (1) using a 0.05 significance level. (8%)
(3) Find and interpret the P-value for this test. (5%)
(4) Construct a 95% confidence interval for p . (6)
(5) What is the probability of committing a type II error when p = 0.4 ? (8%)
A tax assessor wants to estimate the mean property tax bill for all homeowners in a
small town. A survey ten years ago got a sample mean and standard deviation of $1400
and $1000, respectively. (20%)
(1) How many tax records should the tax processor randomly sample in order for a
95% confidence interval for the mean to have a margin error equal to $100? (6%)
(2) In reality, suppose that they’d now got a standard deviation equal to $1500. Using
the sample size you derived in part (1), please find the margin of error. Explain
whether and why this margin of error would be less $100, equal to $100, or more
than $100. (7%)
(3) Refer to part (2). What is the probability that the sample mean falls within $100 of
the population mean? Explain whether and why this probability would be less 0.95,
equal to 0.95, or more than 0.95. (7%)
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