# A series of questions on the topic of probability and statistics.

1.Define the term "hypothesis"; what is a hypothesis test?

2. List and describe two types of error that can occur in hypothesis testing.

3. An independent testing laboratory has been commissioned to determine the relative fuel economy of cars, light trucks, and sport utility vehicles. A random sample of 200 within each category will be taken and their overall miles per gallon will be measured on a controlled course at different speeds. Discuss the nature of the hypothesis on the basis of the type of data (quantitative or qualitative) that is being analyzed and how the population(s) can be compared.

4. List the five steps required to test a hypothesis of the mean. Be sure that all steps are in the correct sequence.

5. The employees of Jones Construction Co. are allowed a 30-minute lunch break. The owner wants to know if the employees are actually taking 30-minute breaks. The foreman decides to record the times taken for the next 25 breaks taken at the site. The data are as follows:

25 34 28 27 33 40 23 22 26

33 38 25 22 27 30 32 31 24

22 28 32 34 26 24 35

a. State the appropriate null and alternative hypotheses.

b. Calculate the test statistic.

c. At p<= 0.05, should the foreman reject the null hypothesis?

d. What conclusion can the foreman present to the owner?

6. The breaking strengths of cables made by a manufacturer have a mean of 1800 pounds and a standard deviation of 100 lb. A new technique is claimed to increase the strength. To test the claim, a sample of 50 cables is tested and it is found that the mean breaking strength is 1850lbs. Can we prove the claim at the .01 significance level?

7. The following table shows the observed frequencies in tossing a die 120times. Test the hypothesis that the die is fair, using a significance level of .05

Die Face 1 2 3 4 5 6

Observed Frequencies 25 17 15 23 24 16

8. As part of a survey on the use of Office Suite Software, the company doing the polling wanted to know whether its population was uniformly distributed over the following age groups: under age 25, 25-44, 45 and over. The company collected the following data:

Age Group Number of respondents

Under 25 73

25-44 61

45 and over 66

(a) Set up the hypotheses to test whether the data are uniformly distributed over the age categories

(b) Find the expected frequency distribution and perform the chi-square goodness of fit test

(c) At the .05 level of significance, would you say that the respondents were uniformly distributed over the age groups?

9. Does a heavier bowler knock down more pins? Jim Samuels, manager of Maple Lanes, asked 15 men's league players to report their body weight (lbs) and their league average, rounded to the nearest pin. Results are shown below:

Weight, X Average, Y Weight, X Average, Y

154 156 215 159

159 145 226 152

165 178 237 132

172 201 245 178

178 205 262 196

192 191 280 145

205 187 289 168

211 146

a. Construct a scatter plot of the data

b. Find the equation for the least squares line of best fit

c. Draw the least squares line on the scatter plot

d. Predict the league average for an adult male bowler who weighs 190 lbs.

What can you conclude?

10. The cotton and wool outlook is published monthly by the Economic Research Service, U.S. Department of Agriculture. The number of 480-lb bales is shown below for 1995 and 1996

1995 1996

Jan 29.94 28.89

Feb 30.14 28.92

Mar 30.72 28.75

Apr 29.82 28.87

May 27.54 28.91

Jun 27.42 33.34

Jul 29.45 33.78

Aug 29.84 34.33

Sep 29.68 34.95

Oct 29.50 3559

Nov 29.37 35.63

Dec 29.18 36.16

(a) use a simple 6-period simple moving average to predict the number of bales for January, 1997

(b) use a weighted moving average to predict the number of bales for January 1997. Use wt-1=.5, wt-2=.3, wt-3=.1, wt-4=.05, wt-5=.03, wt-6=.02

11.. The players on last year's football team at State College were able to bench press a mean of 312 lb. Coach Juarez made it clear to the players during spring training that the team's average best lift had to improve. A special weight-training program was launched, and all the players participated. In an effort to measure the team's progress, the coach recorded the heaviest lifts of the starting offensive and defensive lineups at the start of this season. Results are as follows:

346 412 332 285 396 461 321 275

246 315 298 347 430 419 406 311

319 385 377 365 385 400

a. State the appropriate null and alternative hypotheses.

b. Calculate the test statistic.

c. At  = 0.01, should Coach Juarez reject the null hypothesis?

d. Assuming the starting lineup is a representative sample, what conclusion can the coach draw?

12 The price for purchasing a half-gallon of ice cream in a typical U.S. city is one of many time series collected by the Bureau of Labor Statistics. The monthly prices for 1995 and 1996 are shown below::

1995 1996

Jan 2.650 2.665

Feb 2.55 2.673

Mar 2.686 2.752

Apr 2.629 2.728

May 2.634 2,825

Jun 2.649 2.827

Jul 2.665 2.851

Aug 2.677 2.966

Sep 2.683 3.041

Oct 2.735 3.077

Nov 2.61 2.978

Dec 2.675 2.94

(a) Find the 3-period, 4-period and 5-period Moving Average forecast for January 1997

13. A diaper company is considering 3 different filler materials for their disposable diapers. Eight diapers were tested with each of the 3 filler materials and 24 toddlers were randomly given a diaper to wear. As the child played, fluid was injected into to diaper every 10 minutes until the product failed (leaked). The amount of fluid (in grams) at the time of failure was recorded for each diaper. The data are shown below:

Material 1 Material 2 Material 3

791 809 828

789 818 814

796 803 855

802 781 844

810 813 847

790 808 848

800 805 836

790 811 873

a). What is the response variable and what is the factor?

b) How many levels of the factor are being studies?

c) Is there any difference in the amount of fluid the diaper can hold using the 3 different materials? If so, which ones are different?

d) What is your recommendation to the company?

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Stats

1.Define the term "hypothesis"; what is a hypothesis test?

Solution:

Ill explain the term hypothesis with an example:

Suppose a prospective business person wants to open a new business and the local chamber of commerce office cities the decade old data to claim that the mean household income in the business district is $ 30,000.Should the business person accept this claim?

A hypothesis is a claim or a statement regarding a population parameter which may or may not be true, but needs to be verified by a random sample.

Hypothesis testing is the procedure whereby we decide, on the basis of information taken from a sample, whether to accept or reject a hypothesis.

2. List and describe two types of error that can occur in hypothesis testing.

We accept the null hypothesis when in fact it is true. This is an incorrect decision called Type I error

we accept the null hypothesis when in fact the alternative hypothesis is true. This is an incorrect decision called Type II error.

3. An independent testing laboratory has been commissioned to determine the relative fuel economy of cars, light trucks, and sport utility vehicles. A random sample of 200 within each category will be taken and their overall miles per gallon will be measured on a controlled course at different speeds. Discuss the nature of the hypothesis on the basis of the type of data (quantitative or qualitative) that is being analyzed and how the population(s) can be compared.

Solution:

The type of data is quantitative. The population can be compared based on the 200 samples.

The hypothesis can set as there is no significant relationship in the relative fuel of vehicles.

4. List the five steps required to test a hypothesis of the mean. Be sure that all steps are in the correct sequence.

Solution:

Step 1: State hypotheses.

Step 2: Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.

Step 3: Calculate the test statistic

Step 4: Evaluate the statistic

Step 5: Interpret the result

5. The employees of Jones Construction Co. are allowed a 30-minute lunch break. The owner wants to know if the employees are actually taking 30-minute breaks. The foreman decides to record the times taken for the next 25 breaks taken at the site. The data are as follows:

25 34 28 27 33 40 23 22 26

33 38 25 22 27 30 32 31 24

22 28 32 34 26 24 35

a. State the appropriate null and alternative hypotheses.

b. Calculate the test statistic.

c. = 0.05, should the foreman reject the null hypothesis?At

d. What conclusion can the foreman present to the owner?

Solution:

a. Null Hypothesis:

H0:  = 30

Alternative Hypothesis:

H1:  ≠30

b.

Since the sample size is (n = 25 < 30) small, we can test the above hypothesis by the use of small sample test (t-test) as follows:

Test Statistic:

t =

We obtain the result using megastat

Hypothesis Test: Mean vs. Hypothesized Value

30.000 hypothesized value

28.840 mean Data

5.145 std. dev.

1.029 std. error

25 n

24 df

-1.13 t

.2708 p-value (two-tailed)

c. At α=0.05, the foreman should not reject the null hypothesis, since the p value is > 0.05

d. The foreman present to the owner that the employees are actually taking a 30min break

6. The breaking strengths of cables made by a manufacturer have a mean of 1800 pounds and a standard deviation of 100 lb. A new technique is ...

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Solution to all given problems.