10.2.9

An article in Radio Engineering and Electronic Physics (1980, Vol. 25, pp. 74-79) investigated the behavior of a stochastic generator in the presence of external noise. The number of periods was measured in a sample of 100 trains for each of two different levels of noise voltage, 100 and 150 mV. For 100 mV, the mean number of periods in a train was 7.9 with s₁ = 2.6. For 150 mV, the mean was 6.9 with s₂ = 2.4.

Use α = 0.01 and assume that each population is normally distributed and the two population variances are equal.

(a) It was originally suspected that raising noise voltage would reduce mean number of periods. Do the data support this claim?Choose the answer from the menu in accordance to the question statementChoose the answer from the menu in accordance to the question statement

No.Yes.

(b) Calculate a confidence interval to answer the question in part (a).

μ1−μ2≥Enter your answer in accordance to the question statementEnter your answer in accordance to the question statement. Round your answer to three decimal places (e.g. 98.765).

A local SME bank provides 4 types of loans of its customers and these loans yield the following interest rates to the Bank:

1. Personal Loan 1:         14%
2. Personal Loan 2:         20%
3. Home Loan:                20%
4. Overdraft:                   10%

The Bank has a maximum foreseeable lending capability of Rs 700 million and is further constrained by the policies:

1. Personal Loan 1 must be at least 45% of all personal loans issued and at least 15% of all loans issued (in Rs terms);
2. Personal Loan 2 cannot exceed 40% of all loans issued (in Rs Terms);
3. To avoid public displeasure and the introduction of a new tax, the average interest rate on all loans must not exceed 10%.

Formulate the Linear Programming Problem.

Note: To maximise interest income whilst satisfying the policy limitations.

n chips manufactured, two of which are defective. k chips randomly selected from n for testing.

Q1. What is Pr(a defective chip is in k selected chips) ?

n persons at a party throw hats in a pile, select at random.

Q2. What is Pr(no one gets own hat) ?

Q3. Plot Pr (no one gets own hat) in the Y-axis and n=[1,1000] in the X-axis (~pmf)

Suppose that the IQ of adults is normally distributed with a mean of 100 and standard deviation of 15. not sure with current solution posted, personally it wasn't clear step by step working. I get lost with some values that he gets ( not sure where he gets them )

here's the question:

To get full marks for the following questions you need to convert the question from words to a mathematical expression (i.e. use mathematical notation), defining your random variables where necessary, and using correct probability statements.

Suppose that the IQ of adults is normally distributed with a mean of 100 and standard deviation of 15.

1. (a) [2 marks] What IQ score distinguishes the highest 10%?

2. (b) [3 marks] What is the probability that a randomly selected person has an IQ score between

   91 and 118?

3. (c) [2 marks] Suppose people with IQ scores above 125 are eligible to join a high-IQ club. Show that approximately 4.78% of people have an IQ score high enough to be admitted to this particular club.

4. (d) [4 marks] Let X be the number of people in a random sample of 25 who have an IQ score high enough to join the high-IQ club. What probability distribution does X follow? Justify your answer.

5. (e) [2 marks] Using the probability distribution from part (d), find the probability that at least 2 people in the random sample of 25 have IQ scores high enough to join the high-IQ club.

6. (f) [3 marks] Let L be the amount of time (in minutes) it takes a randomly selected applicant to complete an IQ test. Suppose L follows a uniform distribution from 30 to 60. What is the probability that the applicant will finish the test in less than 45 minutes?

13. The following table identifies a group of children by one of four hair colors, and by type of hair.

• 
  part (b)

• What is the probability that a randomly selected child will have wavy hair? (Enter your answer as a fraction.)

• Part (c)
  What is the probability that a randomly selected child will have either brown or blond hair? (Enter your answer as a fraction.)

• Part (d)
  What is the probability that a randomly selected child will have wavy brown hair? (Enter your answer as a fraction.)

• Part (e)
  What is the probability that a randomly selected child will have red hair, given that he has straight hair? (Enter your answer as a fraction.)

• Part (f)
  If B is the event of a child having brown hair, find the probability of the complement of B. (Enter your answer as a fraction.)

• Part (g)
  If B is the event of a child having brown hair, what does the complement of B represent?

• The complement of B would be the event of a child having blond or black hair.

• The complement of B would be the event of a child having wavy or straight hair.    

• The complement of B would be the event of a child not having brown hair.

• The complement of B would be the event of a child having blond hair.

The business faculty of a public university recorded data on the number of students enrolled in the different study majors for the years 2018 and 2019. These data are useful for the faculty for their decision making process with regard to future planning. The data are stored in BUSSTUDYMAJOR worksheet in the .xls file attached.

1. Use an appropriate graphical technique or chart to display the percentage value of the number of enrolment of the different study major in 2018. Display the chart.

2. Use an appropriate graphical technique or chart to compare the number of enrolment in 2018 and 2019 of the different study major. Display the chart.   

7. Consider the following scenario:

• Let P(C) = 0.2

• Let P(D) = 0.3

• Let P(C | D) = 0.4

• Part (a)
  Find P(C AND D).

• Part (b)
  Are C and D mutually exclusive? Why or why not?C and D are not mutually exclusive because
  P(C) + P(D) ≠ 1
  .C and D are mutually exclusive because they have different probabilities. C and D are not mutually exclusive because
  P(C AND D) ≠ 0
  .There is not enough information to determine if C and D are mutually exclusive.

• Part (c)
  Are C and D independent events? Why or why not?The events are not independent because the sum of the events is less than 1.The events are not independent because
  P(C) × P(D) ≠ P(C | D)
  . The events are not independent because
  P(C | D) ≠ P(C)
  .The events are independent because they are mutually exclusive.

• Part (d)
  Find P(D | C).

8. G and H are mutually exclusive events.

• P(G) = 0.5

• P(H) = 0.3

• Part (a)
  Explain why the following statement MUST be false:
  P(H | G) = 0.4.
  The events are mutually exclusive, which means they can be added together, and the sum is not 0.4.The statement is false because P(H | G) =

• = 0.6. To find conditional probability, divide
  P(G AND H) by P(H)
  , which gives 0.5.The events are mutually exclusive, which makes
  P(H AND G) = 0
  ; therefore,
  P(H | G) = 0.

• Part (b)
  Find
  P(H OR G).

• Part (c)
  Are G and H independent or dependent events? Explain

• G and H are dependent events because they are mutually exclusive.

• G and H are dependent events because

• P(G OR H) ≠ 1.

• G and H are independent events because they are mutually exclusive.

• There is not enough information to determine if G and H are independent or dependent events.

9.

Approximately 281,000,000 people over age five live in the United States. Of these people, 55,000,000 speak a language other than English at home. Of those who speak another language at home, 62.3 percent speak Spanish.

• E = speaks English at home

• E' = speaks another language at home

• S = speaks Spanish at home

Finish each probability statement by matching the correct answer.

• Part (a)
  P(E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (b)
  P(E)
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (c)
  P(S and E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (d)
  P(S | E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

“I don’t like statistical tests,” Ben says grumpily. “There’s so much chance for error, especially Type I error. I don’t understand why we don’t just set alpha really low, like 0.01 or even zero, and have a much lower chance of error.”

What would you say to Ben, in 200 words or less?

4. Assume that the repetition is not allowed. How many orders that can be build from 6 numbers, 2, 3, 4, 5, 7 and 9, taken 3 at a time, to produce any number in hundreds,

a) Without any condition

b) Produce values less than 400.

c) Produce the even values.

d) Produce the odd values.

e) Produce values that are the multiple of 5.

Consider the relationship between hourly wage rate and education attainment. A random sample of 21 male workers was collected to estimate the following model

Yi =β0+β1Xi+ui,fori=1,...,21.

Here,Yi isthelogarithmofhourlywagerate,log(wage),forthei-thworker.Xi istheeducation level, husedu, of the i-th worker, which is measured as the years of schooling, and ui is the error term for the i-th worker. The ordinary least squares (OLS) estimation of the model is reported in the table below. The variable ones

18. (3points) Accordingtotheestimates,whatisthepredictedvalueofthelogarithmofhourly wage for a male worker with 12 years of schooling?

19. (3 points) The standard error of β1 is se(β1) = 0.016 as shown in the OLS results. Test the hypothesis

H0 :β1 =0.03 H1 : β1 ̸= 0.03.

Use the 0.01 significance level.

20. (3 points) Construct the 95% confidence interval for the population parameter β1.

21. (3 points) The standard error of the regression is 0.1853 (not shown in the results table), and the R -squared is 0.525. Use these values to obtain the total sum of squares of the dependent variable.

??

A researcher would like to determine if relaxation training will affect the number of headaches for chronic headache sufferers. For a week prior to training, each participant records the number of headaches suffered. Participants then receive relaxation training and for the week following training the number of headaches is again measured. (18 points) The data are as follows: MD = 2.25, s = 1.28

a)      What is the null hypothesis for this experiment?

b)      What is the alternative hypothesis for this experiment?

c)      Calculate the appropriate test statistic to test the hypothesis- show your work

d)      What is the critical value if alpha is set to .05?

e)      What is the conclusion? State in words whether we reject or fail to reject the null hypothesis and why.

For the data set below, find the upper outlier boundary.
154 160 146 131 148 164 199 169 139 165

Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body bone mineral content (TBBMC). A highly skilled operator is required to take the measurements. Recently, a new DXA machine was purchased by a research lab and two operators were trained to take the measurements. TBBMC for eight subjects was measured by both operators. The units are grams (g). A comparison of the means for the two operators provides a check on the training they received and allows us to determine if one of the operators is producing measurements that are consistently higher than the other. Here are the data:

(a) Take the difference between the TBBMC recorded for Operator 1 and the TBBMC for Operator 2. (Use Operator 1 minus Operator 2. Round your answers to four decimal places.)

Describe the distribution of these differences using words.

(b) Use a significance test to examine the null hypothesis that the two operators have the same mean. Give the test statistic. (Round your answer to three decimal places.)
t =  

Give the degrees of freedom.


Give the P-value. (Round your answer to four decimal places.)


Give your conclusion.

We can reject H₀ based on this sample or We cannot reject H₀ based on this sample.    

(c) The sample here is rather small, so we may not have much power to detect differences of interest. Use a 95% confidence interval to provide a range of differences that are compatible with these data. (Round your answers to four decimal places.

(d) The eight subjects used for this comparison were not a random sample. In fact, they were friends of the researchers whose ages and weights were similar to the types of people who would be measured with this DXA. Comment on the appropriateness of this procedure for selecting a sample, and discuss any consequences regarding the interpretation of the significance testing and confidence interval results.

The subjects from this sample, test results, and confidence interval are representative of future subjects. OR

The subjects from this sample may be representative of future subjects, but the test results and confidence interval are suspect because this is not a random sample