Download QMTH 206 Quiz # 2

QMTH 206
Quiz # 2 - Solutions
Name___________________________
1. Daily sales of a small clothing store are known to follow a normal distribution N( μ, σ ) with a mean = μ and a standard
deviation = σ. Assume that σ is known to have a value of $700. A sample of sales (in dollars only) is taken for 5 days and the
sample mean is computed to be $4500.
a.
What is the random variable in this problem? What are its possible values?
b. What is your best estimate (point estimate) of the population mean?
c.
Compute the 90% confidence interval estimate of the population mean.
a.
X = the amount of sales (in $) in one day at the store.
µ = the average or mean amount of sales for all days at the store.
b.
The best estimate of the population mean µ is the sample mean x-bar which is given at $4500.
c.
The 90% C.I. for µ based on this sample has endpoints
xbar + zα/2 * (σ/√n) = $4500 + zα/2 * ($700/√5)
= $4500 + (1.645)(313.05)
= $4500 + 514.97
= ($3985.03, $5014.97)
is the 90% C.I for the average daily sales based on this sample.
| and since
1 - α = 0.90
| then
α = 0.10
| so
α/2 = 0.05 = 0.0500
| and the table entry with table value 0.0500 is
|
| .04
| (.045) | .05
|
-1.6 | 0.0505 | .0500 | .0495
| then choose zα/2 = 1.645
2. The Environmental Protection Agency (EPA) estimated that the 2003 G-car obtains a mean of 28 miles per gallon on the
highway, and the company that manufactures the car claims that it exceeds the EPA estimate for highway driving. To support its
assertion, the company randomly selected 40 G-cars and recorded the mileage obtained for each car over a driving course similar
to that used by the EPA. This sample had a sample mean = 31.8 mpg.
The EPA assumes that the mileage for these cars is normally distributed and has a standard deviation = 9.0 mpg.
Perform the seven steps to test the hypothesis that the company's claim is correct using a 10% level of significance.
1.
X = mileage for a 2003 G-car.
µ = the average or mean mileage for all 20030 G-cars.
2.
The setup of the hypotheses is
3.
From the random sample of n = 40 cars, the sample mean is xbar = 31.8 mpg.
It is assumed that the population standard deviation is σ = 9.0 mpg.
4.
The formula for the z-test statistic is
zTS = (xbar - µo)/(σ/√n)
= ( 31.8 - 28 ) / ( 9.0/√40)
= ( 3.8 ) / (1.4230...)
= 2.670...
5.
For the level of significance α = 0.10,
then the critical z-value for this test is zα = the table entry for the table value 0.1000
The table values below and above 0.1000 are
| .08 |
| .09
-1.2 | .1003 | .1000 | .0985
and so choose zα = 1.28
6.
The rejection criterion for this upper tail setup of the hypotheses is
Ho : µ < 28 vs. Ha : µ > 28
which is an Upper Tail (UT) setup
Reject Ho if zTS > zα
= 2.67 which is greater than zα = 1.28, then reject Ho and decide that Ha is true.
Since zTS
7.
Conclude that the average mileage for all of the 2003 G-cars is greater than the EPA estimate of 28 mpg, BASED ON
THIS SAMPLE (BOTS).