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# Hypothesis, Probability, Control Charts

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Problem 1)
Identify each of the random variables as continuous or discrete.
a) The number of cows in a pasture.
b) The number of electrons in a molecule.
c) The voltage on a power line.
d) The volume of milk given by a cow per milking.
e) The distance from Caper Canaveral to a point chosen at random in the Sea of Tranquility on the moon.

Problem 2)
At Cape College the business students run an investment club. Each fall they create investment portfolios in multiples of \$1,000 each. Records from the past several years show the following probabilities of profits (rounded to the nearest \$50). In the table below, x = profit per \$1,000 and P(x) is the probability of earning that profit.
x 0 50 100 150 200
P(x) 0.15 0.35 0.25 0.20 0.05

a) Find the expected value of the profit in a \$1,000 portfolio.
b) Find the standard deviation of the profit.
c) What is the probability of a profit of \$150 or more in a \$1,000 portfolio?

Problem 3)
The college health center did a campus-wide survey of students and found that 15% of the students smoke cigarettes. A group of nine students randomly come together and sit at the same table on the plaza in front of the library. Find the probability that:
a) No student at the table smokes.
b) At least one student at the table smokes.
c) More than two students smoke.
d) From one to five smoke (including one and five).

Problem 4)
The probability that a theater patron will request seating on the main floor is 0.35. A random sample of 6 patrons call for tickets. Let r be the number who request main-floor seating.
a) Find P(r) for r=0,1,2,3,4,5, and 6
b) Make a histogram for the r probability distribution.
c) What is the expected number out of 6 who will request main-floor seating?
d) Find the standard deviation of r.

Problem 5)
The Highway Patrol has a target of 28 traffic tickets per week with standard deviation 5 tickets per week for a stretch of mountain highway. The number of tickets issued for 15 consecutive weeks is given below. t = number of week
x = number of tickets
t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
x 22 20 15 16 25 30 30 34 30 32 36 40 33 35 30

a) Make a control chart for the above data.
b) Determine whether the process is in statistical control. If it is not, specify which out-of-control signals are present.

Problem 6)
Jan earned 86 on her political science midterm and 82 on her chemistry midterm. In the political science class the mean score was 80 with standard deviation 4. In the chemistry class the mean score was 70 with standard deviation 6.
a) Convert each midterm score to a standard z score.
b) On which test did she do better compared to the rest of the class?

Problem 7)
The snow pack on the summit of Wolf Creek Pass, Colorado on March 1 has been measured for many years. It is normally distributed with mean 78.1 inches and standard deviation 10.4 inches. A year is selected at random.
a) Find the probability that the snowpack is less than 60 inches.
b) Find the probability that the snowpack is more than 85 inches.
c) Find the probability that the snowpack is between 60 inches and 85 inches.

Problem 8)
Researchers at a pharmaceutical company have found that the effective time duration of a safe dosage of a pain relief drug is normally distributed with mean 2 hours and standard deviation 0.3 hour. For a patient selected at random:
a) What is the probability that the drug will be effective for 2 hours or less?
b) What is the probability that the drug will be effective for 1 hour or less?
c) What is the probability that the drug will be effective for 3 hours or more?

Problem 9)
Life spans for red foxes follow an approximately normal distribution with mean 7 yeas and standard deviation 3.5 years.
a) Find the probability that a red fox chosen at random will live at least 10 years.
b) Find the probability that a random sample of 4 red foxes will have a sample mean life span of over 10 years.

Problem 10)
Margot has found that the mean time to run a job at the college copy center is 12.6 minutes with standard deviation 10 minutes. She selects a random sample of 64 jobs.
a) What is the probability that the sample mean copying time is between 14 and 15 minutes?
b) What is the probability that the sample mean copying time for this sample is between 10 and 12 minutes?

Problem 11)
A random sample of 50 calls initiated on cellular car phones had a mean duration of 3.5 minutes with standard deviation 1.2 minutes. Find a 99% confidence interval for the population mean duration of telephone calls initiated on cellular car phones.

Problem 12)
A park ranger has timed the mating calls of 22 bull elk. The mean duration of these calls is 14.6 seconds with standard deviation 2.8 seconds. Find a 95% confidence interval for the population mean duration of mating calls of the bull elk.

Problem 13)
The heights of a random sample of six fifth-graders are the following (in inches):
49.4 59.5 50.8 55.0 60.1 53.3
Compute a 90% confidence interval for all fifth-graders from this data.

Problem 14)
Jerry is doing a project for his sociology class in which he tests the claim that the Pleasant View housing project contains family units of average size 3.3 people (the national average). A random sample of 64 families from Pleasant View project shows a sample mean of 4.3 people per family unit with sample standard deviation 1.3. Construct a hypothesis test to determine whether the average size of a family unit in Pleasant View is different from the national average of 3.3. Use a 5% level of significance.

a. State the null and the alternate hypothesis.
b. Identify the sampling distribution to be used: the standard normal distribution or the Student's t distribution. Find the critical values.
c. Compute the z or t value of the sample test statistic.
d. Find the P value or an interval containing the P value for the sample test statistic.
e. Based on your answers to a through d, decide whether or not to reject the null hypothesis at the given significance level. Explain your conclusion in simple, nontechnical terms.

Problem 15)
Statistical Abstracts (117th edition) reports that the average amount spent annually for food by householders under 25 years of age is \$2,690. A random sample of 16 people under 25 years of age who live in a university neighborhood were surveyed. The survey showed that they spend a sample mean \$3,220 with sample standard deviation \$750. Test the claim that the mean for this neighborhood is greater than the national average. Use a 5% significance level.

a. State the null and the alternate hypothesis.
b. Identify the sampling distribution to be used: the standard normal distribution or the Student's t distribution. Find the critical values.
c. Compute the z or t value of the sample test statistic.
d. Find the P value or an interval containing the P value for the sample test statistic.
e. Based on your answers to a through d, decide whether or not to reject the null hypothesis at the given significance level. Explain your conclusion in simple, nontechnical terms.

Problem 16)
The board of real estate developers claims that 55% of all voters will vote for a bond issue to constructor a massive new water project. A random sample of 215 voters was taken and 96 said that they would vote for the new water project. Test to see if this data indicates that less than 55% of all voters favor the project. Use a 1% significance level.

a. State the null and the alternate hypothesis.
b. Identify the sampling distribution to be used: the standard normal distribution or the Student's t distribution. Find the critical values.
c. Compute the z or t value of the sample test statistic.
d. Find the P value or an interval containing the P value for the sample test statistic.
e. Based on your answers to a through d, decide whether or not to reject the null hypothesis at the given significance level. Explain your conclusion in simple, nontechnical terms.

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