Quiz 3
1. (5 points) What is the null hypothesis?
2. (5 points) What is the alternative hypothesis?
3. (5 points) What is a type I error? Be sure to explain α, the significance level, and (1α) the confidence level.
4. (5 points) What is a type II error? Be sure to explain β, which has no standard name, and (1β), the power.
5. (10 points) The distribution of weights for a large sack of rice is approximately a mean of 100 lbs and a standard deviation of 2 lbs.
a. Use the ztable calculator and find weight of a sack which is in the 99th percentile. Use value from an area. (No screen shot necessary.)
http://davidmlane.com/hyperstat/z_table.html
b. What is the zscore for this weight? (zscore = (xµ)/σ)
c. Suppose you want to try to save money. You will weigh each sack first and then only buy the sack if it weighs 101 lbs or more. If you weigh 100 sacks, how many of these sacks will you end up buying on average? (Show a screen shot of your answer.)
Use the central limit theorem to answer the following questions.
d. Suppose instead to save time, you decide to weigh 10 sacks at a time and take the average value. What is the average weight of a sack? (µ)
e. For a group of 10 sacks, what would be the standard deviation of the group? (i.e. σ/)
f. If you only buy the group of 10 sacks when its average weight is 101 lbs or more, what is the percentage of the groups of 10 that you will end up buying? Use the ztable calculator. (No screen shot necessary)
6. (10 points) Download the data for the Quiz 1 Excel data file. Use Column A.
A certain drug is said to produce at least 3.4 micrograms of its product in the bloodstream. We want to test whether the data taken from 10 test patients shows this effect (µ >= 3.4). If the effect is less than its advertised level, then we will report the company for false advertising.
a. What is the null hypothesis H_{o} for our test?
b. What is the alternative hypothesis H_{a}?
c. What type of tail test will we use? (left tail, right tail, or two tails)?
d. What is the mean of the sample xbar?
e. What is the standard deviation of the sample s?
f. What is the size of the sample n?
g. We going to use a tstatistic. Explain why we are not going to use a zstatistic.
h. Calculate the tstatistic using xbar, µ, n, and s.
i. How many degrees of freedom does this data set have?
j. Use the tdistribution calculator to compute a pvalue. Include a screen shot of your answer.
http://www.statdistributions.com/t/
k. Typically, we want a 95% confidence level. Based on your value of p, should we accept or reject the null hypothesis?
l. What if we want a 99% confidence level? Based on your value of p, should we accept or reject the null hypothesis?
7. (10 points) Use Columns C and D for this question.
You are a manager and are testing two methods of production – Method C and Method D. You want to know whether there is any difference in the output depending on which method is used. Column C gives the output for 9 workers using the first method. Column D gives the output for the SAME 9 workers using the second method. Now there are two ways to do the problem. We could test if µ(C) is equal to µ(D). Or since there is data for the same workers using different methods, we could test whether µ(CD) = 0.
The easier way to do this problem is to test whether µ(CD) equals 0, or is not equal to 0.
a. Make a new series of data samples by letting E = C – D. List your new series of 9 numbers.
b. What is the null hypothesis H_{0} ?
c. What is the alternative hypothesis H_{a} ?
d. What type of tail test are we going to use? (left tail, right tail, two tail)
e. What is the mean xbar of this new sample?
f. What is the standard deviation of the sample s?
g. What is the size of the sample n?
h. How many degrees of freedom does this data set have?
i. What is the tstatistic for this sample?
j. Use the tdistribution calculator to compute a pvalue. Show a screen shot of your answer.
k. Based on this value of p and using a 95% confidence level, is there a difference between the two methods in the production? Should we accept or reject the null hypothesis?
8. (10 points) Use columns F and G for the LeastSquares line.
a. Use Excel to make a scatter plot of the data.
b. Adjust the values of the x and y axes so that the data is centered in the plot.
c. Put the trendline on your plot.
d. Put the equation of the trendline on your plot.
e. Put the R^{2} value on your plot.
f. The R value is a measure of how well the data fits a line. What is R?
g. Make a screen shot of your final plot. How well do you think the data fits the line? (good fit, moderate fit, marginal fit, no fit)
9. (10 points) You are working for the Center for Disease Control (CDC). It is flu season and you suspect that flu is affecting the very young and the very old more than other age groups. Your hypothesis is that the number of sick people in each age group should be about the same. You are going to use a Xsquared test on your hypothesis. The average numbers of people who were reported to have the flu last month was:
0 – 10 years – 185
11 – 20 years – 175
21 – 30 years 140
31 – 40 years – 145
41 – 50 years – 150
51 – 60 years –140
61 – 70 years – 165
71 and over – 180
a. What is the null hypothesis?
b. What is the alternative hypothesis?
c. What is the total number of people who were reported sick last month?
d. Your model is that the number of people who have the flu should be the same in each age group. Therefore, what is the expected number of people who should be sick in each age group?
e. Enter the observed number of people who have the flu and the expected number of people who have the flu into the Xsquared goodness of fit applet.
http://home.ubalt.edu/ntsbarsh/Businessstat/otherapplets/goodness.htm
f. What is the number of degrees of freedom?
g. What is the pvalue? Provide a screen shot of your answer.
h. Using a 95% confidence interval, should you accept or reject the null hypothesis?
10. (10 points) This problem is the check to see whether you understand the Xsquared test. There are only 2 test columns, so you cannot use the Xsquared Goodness of Fit applet from the previous problem as it requires 3 or more test intervals.
You are flipping coins and wonder if the coin is “fair” or “weighted”. After 100 flips, you get 60 heads and 40 tails. Determine if the coin is “fair” or “weighted” using a Xsquared test.
a. What is the null hypothesis for this test?
b. What is the alternative hypothesis?
c. Fill in the following table.
Flip

Observed

Expected

(O – E)

(OE)^{2}

(OE)^{2}/E

Heads

60





Tails

40





Sum

100



n/a


d. What is the value of X^{2} for this data?
e. What is the number of degrees of freedom?
f. Use the X^{2} calculator to compute p (use the right tail option). Provide a screen shot of your calculation.
http://www.statdistributions.com/chisquare/
g. Does this value of p support the null hypothesis strongly, moderately, weakly? Or does the value of p support the alternative hypothesis? Explain why.
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