5.1 Combination and permutation

a) ₅C₂

b) 5!

c) Five different drugs, A, B, C, D, and E, can be used to treat a disease in different combinations. If a physician uses two of them to treat patients, how many combinations are possible? List all such combinations.

d) Four different exercises, A, B, C, and D, are recommended for an injury recover therapy program. The therapists want to know whether there is a different treatment effect using i) just one; ii) any two; or iii) any three of the therapies. How many treatment regimens can a therapist choose from? What are they?

e) If the order matters in d), if therapists were to select two exercises out of four, how many treatment options would the therapists have? What are they?

A hospital outpatient clinic performs four types of operations. The profit per operation, as well as the minutes of X-ray time and laboratory time used, are given in the table below. The clinic has 500 private rooms and 500 intensive care rooms. Type 1 and type 2 operations require a patient to stay in an intensive care room for one day, whereas type 3 and type 4 operations require a patient to stay in a private room for one day. Each day, the hospital is required to perform at least 100 operations of each type. The hospital has set the following goals (listed in order of priority):

■ Goal 1: Earn a daily profit of at least$100,000.

■ Goal 2: Use at most 50 hours daily of X-ray time.

■ Goal 3: Use at most 40 hours daily of laboratory time. Use goal programming to determine the types of operations that should be performed.

Please conduct a Hypothesis test in the following scenario to determine if the population slope is statistically significant. State the appropriate test statistic, the critical values, and whether or not you can reject a null hypothesis that the population slope is zero.

Use a 1% significance level. Suppose that you collected data to determine the relationship between the amount of time a person spends online as an independent variable and the amount of money a person spends online as the dependent variable.

The regression equation is = 24 + 8.5x, where x represents the number of hours a person spends online and represents the predicted amount of money that person spends online.

Sample size is 6.

SSE is 200

SSR is 800

SST is 1000

Standard deviation of the sampling distribution of the sample intercept is 9.54

Standard deviation of the sampling distribution of the sample slope is 2.26

Agri-Beef, Inc. is a large Midwestern farming operation. The company has been a leader in employing statistical techniques in its business. Recently, John Goldberg, operations manager, requested that a random sample of cattle be selected and that these cattle be fed a special diet. The cattle were weighed before the start of the new feeding program and at the end of the feeding program. John wished to estimate the average daily weight gain for cattle on the new feed program. Two hundred cattle were tested, with the following sample results: X = 1.2 pounds gain per day and S= 0.50 pounds gain per day.

1. Obtain a 90% confidence interval estimate for the true average daily weight gain.

2. Provide a 96% confidence interval estimate for the true average daily weight gain.

3. John is considering adopting this new diet. However, the weight gain comes at a price. To feed 200 cows for one month, the diet would cost approximately $1,000 more than their current feed program. If the price of beef on the hoof has been close to $0.20 a pound, would such a program be cost effective for Agri-Beef? Support your answer with calculations and statistical reasoning.

A researcher claims the mean age of residents of a small town is more than 38 years. The age (in years) of a random sample of 30 sutdents are listed below. At alpha=0.10, is there enough evidence to support the researcher's claim? assum the population standard deviation is 9 years.

Ages (in years)
41
33
47
31
26
39
19
25
23
31
39
36
41
28
33
41
44
40
30
29
46
42
53
21
29
43
46
39
35
33
42
35
43
35
24
21
29
24
25
85
56
82
87
72
31
53
31
33
54
60
31
81
32
40
26
52
37
71

a) Identify the claim and state Ho and Ha
(b) Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed and whether to use a z-test, a t-test, or a chi-square test. Explain your reasoning
(c)Choose one of the options Option 1: Find the critical value(s), identify the rejection region(s), and find the appropriate standardized test statistic. Option 2: Find the appropriate standardized test statistic and the P-value
(d) Decide whether to reject or fail to reject the null hypothesis
(e) Interpret the decision in the context of the original claim.

The average number of acres burned by fires in New Mexico County is assumed to be normally distributed with a mean of 9000 and a standard deviation of 695 acres. You are told that the probability of the number of acres burned between 9350-R and 9350+R is .7802. Find R. (Hint: Draw a picture to see the area along with the given probability. Can Goal-Seek or trial and error help you find the value of R?)

How would I solve for R using goal seek in excel

A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions. (Round your answers up to the nearest integer.)

(a) What sample size is necessary if the 95% CI for p is to have a width of at most 0.12 irrespective of p?

(b) If the legislator has strong reason to believe that at least 3/4 of the electorate know of her position, how large a sample size would you recommend to maintain a width of at most 0.12?

Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma.† Over a period of months, an adult male patient has taken ten blood tests for uric acid. The mean concentration was x = 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with σ = 1.87 mg/dl.

(a) Find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error? (Round your answers to two decimal places.)

(b) What conditions are necessary for your calculations? (Select all that apply.)

uniform distribution of uric acid

σ is known

σ is unknown

n is large

normal distribution of uric acid

(c) Interpret your results in the context of this problem:

There is a 5% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.

There is a 95% chance that the confidence interval is one of the intervals containing the population average uric acid level for this patient.    

The probability that this interval contains the true average uric acid level for this patient is 0.05.

The probability that this interval contains the true average uric acid level for this patient is 0.95.There is not enough information to make an interpretation.

(d) Find the sample size necessary for a 95% confidence level with maximal margin of error E = 1.20 for the mean concentration of uric acid in this patient's blood. (Round your answer up to the nearest whole number.)
_____blood tests

A communication system consists of n antennas of which m are defective and n − m are functional. Suppose the n antennas are indistinguishable and are lined up in a linear array. The system is called functional as long as no two consecutive antennas are defective. Suppose we randomly line up the n antennas. What is the probability that the resulting system will be functional?

A simple random sample of size n is drawn from a population that is normally distributed. The sample​ mean,

x overbarx​,

is found to be

109109​,

and the sample standard​ deviation, s, is found to be

1010.

​(a) Construct

aa

9090​%

confidence interval about

muμ

if the sample​ size, n, is

1515.

​(b) Construct

aa

9090​%

confidence interval about

muμ

if the sample​ size, n, is

2626.

​(c) Construct

aa

9898​%

confidence interval about

muμ

if the sample​ size, n, is

1515.

​(d) Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

A psychology professor assigns letter grades on a test according to the following scheme.

A: Top 10%of scores

B: Scores below the top 10%and above the bottom 55%

C: Scores below the top 45%and above the bottom 20%

D: Scores below the top 80%and above the bottom 7%

F: Bottom 7%of scores

Scores on the test are normally distributed with a mean of 67.6and a standard deviation of 9. Find the numerical limits for a D grade. Round your answers to the nearest whole number, if necessary.

Suppose you are interested in whether the GPAs of college athletes are significantly different from the GPAs of the college athlete population. To investigate this, you obtain a random sample of academic records of 5 athletes and compare their GPA to the known student population.

Here are your data:

Using an alpha = .05, what is your alternative/research hypothesis?

The overhead reach distances of adult females are normally distributed with a mean of 202.5 cm and a standard deviation of 8.9 cm.

a. Find the probability that an individual distance is greater than 211.80 cm.

b. Find the probability that the mean for 20 randomly selected distances is greater than 200.70 cm.

c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?