### In-class Exercises

31st July

1. A manager wishes to determine whether the mean times required to complete a certain task differ for the three levels of employee training. He randomly selected 10 employees with each of the three levels of training (Beginner, Intermediate and Advanced). Do the data provide sufficient evidence to indicate that the mean times required to complete a certain task differ for at least two of the three levels of training? The data is summarized in the table.

 Level of Training n s2 Advanced 10 24.2 21.54 Intermediate 10 27.1 18.64 Beginner 10 30.2 17.76

Ans: Solution is available here - http://www2.fiu.edu/~howellip/exanova.htm

2. The following F-test was generated from a publicly available data set. The data set contains 480 ceramic strength measurements for two batches of material. The summary statistics for each batch are shown below.Test whether the population variances are different or not.

BATCH 1:
NUMBER OF OBSERVATIONS      =      240
MEAN                        =    688.9987
STANDARD DEVIATION          =    65.54909

BATCH 2:
NUMBER OF OBSERVATIONS      =      240
MEAN                        =    611.1559
STANDARD DEVIATION          =    61.85425

3.

A sports psychologist was interested in testing the effect of a simple relaxation technique on college basketball players’ free throw shooting accuracy. Each player was asked to shoot 20 consecutive free throws and the number of successful attempts was recorded. The players were then trained to use a simple 5 second relaxation technique while preparing to shoot a free throw. The players then returned to the court and shot 20 consecutive free throws again. The resulting data are given below. What would be the psychologist's conclusion?

Number of Free Throws Completed Per Twenty Attempts

 pre post 12 13 15 15 9 11 16 15 12 15 15 18 17 17 10 12 12 13 14 17

4. Suppose that you interview 1000 exiting voters about who they voted for governor.  Of the 1000 voters, 550 reported that they voted for the democratic candidate.  Is there sufficient evidence to suggest that the democratic candidate will win the election at the .01 level?

5. A random survey of 80 homeowners in a subdivision of 512 homes found that the average yearly insurance premium on their homes was \$584.50 with a sample deviation of \$64.85. Test the following three claims about the true average premium µ each at the 0.05 level of significance.

 1. Ho: µ <= 600 2. Ho: µ >= 600 3. Ho: µ = 600

24th July

1.

An insurance company is reviewing its current policy rates. When originally setting the rates they
believed that the average claim amount was \$1,800. They are concerned that the true mean is actually
higher than this, because they could potentially lose a lot of money. They randomly select 40 claims, and
calculate a sample mean of \$1,950. Assuming that the standard deviation of claims is \$500,
test to see if the insurance company should be concerned.

2.

A sample of 40 sales receipts from a grocery store has an average of \$137 and standard deviat6ion of \$30.
Use these values to test whether or not the mean is sales at the grocery store are different from \$150.

3.

Mr Brown is the owner of a small bakery in a large town. He
believes that the smell of fresh baking will encourage customers to
purchase goods from his bakery. To investigate this belief, he
records the daily sales for 10 days when all the bakery's windows
are open, and the daily sales for another 10 days when all the
windows are closed. The following sales, in \$, are recorded.

Windows Open

202.0   204.5    207.0     215.5    190.8

215.6   208.8    187.8     204.1    185.7

Windows closed

193.5   192.2   199.4    177.6   205.4

200.6    181.8   169.2    172.2    192.8

Assuming that these data may be deemed to be random samples
from normal populations with the same variance, investigate the
baker's belief.

4.

An insurance company states that 90% of its claims are settled
within 30 days. A consumer group selected a simple random
sample of 75 of the company’s claims to test this statement.
The consumer group found that 55 of the claims were settled
within 30 days. At the 0.05 significance level, test the
company’s claim that 90% of its claims are settled within 30
days.

17th July

1. A population consists of 4 values. If you list all samples of size 2, how many samples would you get? If the mean and standard deviation of the population are 10 and 2 respectively, can you infer anything about the mean and standard deviation of the sample means?  2 + 2 + 2   = 6 points

2.

A normal distribution has a mean of 50 and a standard deviation of 4. Determine the value below which 95% of the observations will occur. What would be the probability of getting an observation which is less than 40?

3   + 3 = 6 points

3.

The amount of cola in a 11-ounce can is uniformly distributed between 10 ounces and 12 ounces. What is the mean amount per can? What is the standard deviation amount per can? What is the probability of selecting a can of cola and finding it has less than 11 ounce? What is the probability of selecting a can of cola and finding it has more than 11 ounce?    2 + 2 + 2 + 2 = 8 points

4. The mean GPA for the BUS 210 students is 3.4 and the standard deviation is 0.4.  The total number of students in the class is 60.

a) If you select a random sample of 36 students from this class, how many samples you can have? 3 points

b) What would be the mean and standard deviation of your sample means? 3 points

c) What is the probability that in one sample, the average GPA would be less than 3.0? 4 points

10th July

1.      You are investing \$1000 in a business which would give you a gross return of - \$5000 with probability 0.5, \$2000 with 0.4 and \$0 with probability 0.1

(i)                 What would be the expected average net return (gross return – investment)? 3 points

(ii)                What would be the expected standard deviation of the net return? 3 points

2.

The mean income of a group of sample observations is \$500; the standard deviation is \$40.

According to Chebyshev’s theorem, at least what percent of the income will lie between \$400 and \$600? (4 points)

3.

Chipper Jones of the Atlanta Braves had the highest batting average in the 2008 Major League Baseball season. His average was 0.364. So, assume the probability of getting a hit was 0.364 for each time he batted. In a particular game assume he batted three times.

1. In 100 innings what would be his expected average?
2. In 100 games what would be his expected average (assume that he batted in all three times in every game he played)?
3. What is the probability of getting at least 2 hits in one game?
4. What is the probability of getting at most 2 hits in one game?
5. What is the probability of getting either 0 hit or 3 hits in one particular game?

9th July

1. The total number of students in East Seattle Community College is 20; 15 of them are female and 5 male. 2 of the male students are enrolled in the Accounting class, 1 in the Economics class and 2 in the Statistics class. 6 of the female students are enrolled in the Accounting class, 4 in the Economics class and 5 in the Statistics class.

Now if you randomly select a student what would be the following probabilities?

(i)                  Probability (Enrolled in the Accounting class)

(ii)                Probability (A male student and enrolled in the Statistics class)

(iii)               Probability (Female student given that the student is enrolled in the Economics class)

(iv)              Probability (Enrolled in the Accounting class given that the student is male)

(v)                Probability (Enrolled in the Economics class and a female student)

2. The probabilities of increase in stock price for Google, Microsoft and Toyota are 0.4, 0.5 and 0.2 respectively. The probability of increase in stock prices for both Google and Microsoft would be 0. 1.  The probability of increase in stock prices for both Google and Toyota would be 0.8

Given these information, evaluate the following statements (explain whether the statements are right or wrong).  (5 X 2 points)  = 10 points

1. Stock price increase for Google and stock price increase for Microsoft are mutually exclusive events.
2. Stock price increase for Google and stock price increase for Toyota are mutually exclusive events.
3. Stock price increase for Google and stock price increase for Microsoft are independent events.
4. Stock price increase for Google and stock price increase for Toyota are independent events.
5. The probability of  increase in stock price for Google or Microsoft would be 0.8

2nd July, 2013

1.

The monthly salaries of a sample of 60 employees were rounded to the nearest ten dollars. They ranged from a low of \$1,040 to a high of \$1,720.

(i) How many classes you must have? Explain.    3 points

(ii) What would be the class interval? Explain.     2 points

(iii) Give one example of feasible classes (clearly mention the lower limit and upper limit of each class).  3 points

2.

Refer to the following breakdown of responses to a survey of room cleanliness in a hotel.

Response           Frequency

Not satisfied      20

Satisfied              40

Highly satisfied  20

What percent of the responses did not indicate that customers were not satisfied?   2 points

3. In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?

4.

 A single 6-sided die is rolled. What is the probability of rolling a number greater than 3 or an even number?

5.
A number from 1 to 10 is chosen at random. What is the probability of choosing a 5 or an even number?

6.
 In a pet store, there are 6 puppies, 9 kittens, 4 gerbils and 7 parakeets. If a pet is chosen at random, what is the probability of choosing a puppy or a parakeet?

8/1/2012

1. The processors of Fries’ Catsup indicate on the label that the bottle contains 16 ounces of catsup.  The standard deviation of the process is 0.5 ounces.  A sample of 36 bottles from last hour’s production revealed a mean weight of 16.12 ounces per bottle.  At the .05 significance level is the process out of control?  That is, can we conclude that the mean amount per bottle is different from 16 ounces?

2. Roder’s Discount Store chain issues its own credit card.  Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than \$400.  The level of significance is set at .05.  A random check of 172 unpaid balances revealed the sample mean to be \$407 and the sample standard deviation to be \$38. Should Lisa conclude that the population mean is greater than \$400, or is it reasonable to assume that the difference of \$7 (\$407-\$400) is due to chance?

3. The current rate for producing 5 amp fuses at Neary Electric Co. is 250 per hour.  A new machine has been purchased and installed that, according to the supplier, will increase the production rate.  The production hours are normally distributed.  A sample of 10 randomly selected hours from last month revealed that the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. At the .05 significance level can Neary conclude that the new machine is faster?

4. In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution.  A new solicitation letter that has been drafted is sent to a sample of 200 people and 45 responded with a contribution.  At the .05 significance level can it be concluded that the new letter is more effective?

7/24/2012

1. Nike’s annual report says that the average American buys 6.5 pairs of sports shoes per year. Suppose the population standard deviation is 2.1 and that a sample of 81 customers will be examined next year.

a.       What is the standard error of the mean in this experiment?

b.      What is the probability that the sample mean is between 6 and 7 pairs of sports shoes?

c.       What is the probability that the difference between the sample mean and the population mean is less than 0.25 pairs?

d.      What is the likelihood the sample mean is greater than 7 pairs?

2. The Roman Arches is an Italian restaurant. The manager wants to estimate the average amount a customer spends on lunch Monday through Friday. A random sample of 15

customers' lunch tabs gave a mean of \$9.74 with a standard deviation of  \$2.93.

(a) Find a 95% confidence interval for the average amount spent on lunch by all customers.

(b) How would your answer change if population Size is given as 500.

(c) How would your answer change if sample size is given as 115.

(d) For a day when the Roman Arches has 115 lunch customers, estimate the range of dollar values for the total lunch income that day.

3. The following information is available.

H0 :  Population Mean = 12.5

The sample mean is 13.4, and the sample size is 51. The population standard deviation is 4.1.

(i) Find the computed z value.

(ii) Find the p value.

(iii) Test the hypothesis at .05 significance level.

7/11/2012

1. In a recent survey 35 percent indicated chocolate was their favorite flavor of ice cream. Suppose we select a sample of 10 people and ask them to name their favorite flavor of ice cream.

1. How many of those in the sample would you expect to name chocolate?
2. What is the probability exactly four of those in the sample name chocolate?
3. What is the probability four or more name chocolate?
4. What is the probability that at most five people would name some other flavor?
5. What is the probability that at least three would name some other flavor?
2.  A federal study reported that 7.5% of the U.S. workforce has a drug problem. A drug enforcement official for the state of Indiana wished to investigate this statement.  In her sample of 20 employed workers:

a. How many would you expect to have a drug problem? What is the standard deviation?

b. What is the likelihood that none of the workers sampled has a drug problem?

c. What is the likelihood at least one has a drug problem?

ĉ
Prabirendra Chatterjee,
Jul 1, 2012, 11:19 PM