Hi,

Working through a chapter on sampling distributions and getting stuck very early on something that must be so simple the book didn't see fit to explain properly.

Example: In a certain population 30% of people are of blood type A. A random sample of size 5 is drawn. Therefore the population proportion with type A is p = 0.3.
The possible values of pˆ are 0, 0.2, 0.4, 0.6, 0.8, 1.

Why are these the possible values of p^? The numbers given create 5 intervals - is it because we have a sample of 5? If the sample was 10, would the possible values changed? It makes sense to me that they would, but my book doesn't go this way, the next paragraph shows that P(0.2 ≤ pˆ ≤ 0.4) increases as the sample size increases so it seems that the 'possible values' haven't moved.

Could anybody explain to me what is the story with the 'possible values' pf p^ here?

Thanks a lot

A car salesman tells you that the model you are looking at gets on average 36 miles per gallon overall (highway and street driving) with a standard deviation of 4 miles per gallon. He also tells you that 20% of the cars get at
least 44 miles per gallon.

a. ( Do you think this is a normal distribution? Explain why or why not. (there is no one right answer for this)

b. Analyze the salesman's statement mathematically. Is he being at all untruthful?

i. Explain under the assumption that the distribution IS a normal distribution.

ii. Explain under the assumption that the distribution is NOT a normal distribution.

According to the U.S. Federal Highway Administration, the mean number of miles driven annually is 12,200 with a standard deviation of 3800 miles. A resident of the state of Montana believes the drivers in Montana drive more than the national average. She obtains a random sample of 35 drivers from a list of registered drivers in the state and finds the mean number of miles driven annually for these drivers to be 12,895.90. Is there sufficient evidence to show that residents of the state of Montana drive more than the national average?

What is the p-value for this hypothesis test?
What is the test statistic for this hypothesis test?
What is the critical value?
What is the decision?

The mean tread life of a manufacturer's best selling tire is known to be 22,560 miles. Because the distribution of individual tire life is positively skewed, it cannot be assumed to be normal. R&D has come up with a new rubber additive that they suspect will increase mean tire life without affecting variability of tire life. A sample of 36 tires with the new additive has a sample mean tread life of 24,470 miles; the sample based estimate of the standard deviation is 1200 miles. At .05 level of significance, test whether the additive is working. Clearly state your hypothesis and conclusion.

The tread life (x)  of tires follow  normal distribution with µ =  60,000 and σ= 6000 miles.  The manufacturer guarantees the tread life for the first 52,000 miles. (i) What proportion of tires last  at least 55,000 miles? (ii)  What proportion of the tires will need to be replaced under warranty? (iii)  If you buy 36 tires,  what is the probability that the  average life of your 36 tires will exceed  61,000?  (iv) The manufacturer is willing to replace only 3% of its tires under a warranty program involving tread life.  Find the tread life covered under the warranty.

What is the NPV for the car selection example below?

Example: Buy an Electric Vehicle?

Chevy Malibu: $30,000 cost  12K miles/yr  30 mph  Maintenance: $1K in year 1; 5% CAGR  Resale of $5K in ten years  Gas: $3.50/g; 10% CAGR  D = 7% 

Nissan Leaf: $40,000 cost  12K miles/yr  5 miles per kWh  Maintenance: $700 in year 1; 5% CAGR  New battery in year 5: $5,000  Resale of $3K in ten years  $0.11/kWh; 10% CAGR  D = 7%

Suppose that the manager of the MileagePlus frequent flier program is promoted and consequently another individual is hired to replace him. Also suppose that United publishes in internal documentation that the average number of Premier Qualifying Miles (PQM) earned by individuals who travel for work at least once a month is 45,000 with a standard deviation of 5,000 miles. Further suppose that the new manager desires to test the claims that United has made to see if the statistics have changed.

a) First, are these statistics given by United describing the parent population or a sample ?

b) Define appropriate null and alternative hypothesis

c) Suppose that the new manager takes a sample of 50 such United customers. What is the probability that a sample of size 50 provides a sample mean within + - 1,000 miles of the 45,000 mile figure provided by United?

d) What is the probability that a sample of 50 such United customers provides a sample mean wiithin + - 500 miles?

e) Suppose that the new manager's sample has a sample average of 47,500 miles. Compute the 95% confidence interval for the population mean.

f) Based on the confidence interval you computed in part e), does this sample provide evidence for or against United's claim that the average number of Premier Qualifying Miles (PQM) earned is 45,000 ?

g) If the new manager wants to determine a specific p-value for the likelihood that the null hypothesis is true based on the sample collected in part c), should he use Z- or t- scores for the test statistic?

h) Determine and interpret the p-value

Bob's Cars purchased a new car for use in its business on January 1, 2017. It paid $30,000 for the car. Charm expects the car to have a useful life of four years with an estimated residual value of zero. Charm expects to drive the car as follows

Year 2017: 20,000 miles driven

Year 2018: 45,000 miles driven

Year 2019: 40,000 miles driven

Year 2020: 15,000 miles driven

Total Miles Driven: 120,000 miles

1. Using the straight-line method of depreciation, calculate the following amounts for the car for each of the four years of its expected life:
a.   Depreciation expense
b.   Accumulated depreciation balance
c.   Book value

2. Using the Units-of-Production method of depreciation, calculate the following amounts for the car for each of the four years of its expected life:
a.   Depreciation expense
b.   Accumulated depreciation balance
c.   Book value

3. Using the Double-Declining-Balance method of depreciation, calculate the following amounts for the car for each of the four years of its expected life:
a.   Depreciation expense
b.   Accumulated depreciation balance
c.   Book value

4. Using the Double-Declining-Balance method of depreciation, calculate the following and record the appropriate journal entry assuming all events occurred at end of 2018.
a.   Assume the car was involved in an accident, damaged beyond repair, and disposed of.
b.   Assume the car was involved in an accident and sold for $1,000.
c.   Assume the car was limited edition and resold for $24,000.

An automobile manufacturer who wishes to advertise that one of its models achieves 30 mpg (miles per gallon) decides to carry out a fuel efficiency test. Six nonprofessional drivers were selected, and each one drove a car from Phoenix to Los Angeles. The resulting fuel efficiencies (in miles per gallon) are given below.

Assuming that fuel efficiency is normally distributed under these circumstances, do the data contradict the claim that true average fuel efficiency is (at least) 30 mpg? Test the appropriate hypotheses at significance level 0.05. (Use a statistical computer package to calculate the P-value. Round your test statistic to two decimal places and your P-value to three decimal places.)

t=P-value=

State the conclusion in the problem context.

Reject H₀. We have convincing evidence that the mean fuel efficiency under these circumstances is less than 30 miles per gallon.Reject H₀. We do not have convincing evidence that the mean fuel efficiency under these circumstances is less than 30 miles per gallon.    Do not reject H₀. We have convincing evidence that the mean fuel efficiency under these circumstances is less than 30 miles per gallon.Do not reject H₀. We do not have convincing evidence that the mean fuel efficiency under these circumstances is less than 30 miles per gallon.

Q5. Find if each Cobb-Douglas production functions below is constant return to scale, increasing return to scale, or decreasing return to scale?

Q=20K^0.8 L^0.5

Q=35K^0.8 L^0.8

Q=40K^0.3 L^0.5