3. Let Y be a singleobservation from a population with density function.

f(y)=  2y/θ2 0 ≤ y ≤ θ  

f(y)=  0     elsewhere

(a) Find the MLE for θ.

(b) Is the MLE found in (a) also a MVUE of θ? Justify your answer.

(c) Show that Y/ θis a pivotal quantity for θ.

(d) Use the pivotal Y /θ to find a 100(1 − α)% upper confidence interval for θ, which is of form (−∞, θˆU ).

For the following questions, suppose for some θ0 > 0, we wish to test H0 :θ = θ0v.s. H1 :θ < θ0 based on a single observation Y .

(e) Show that a level-α test has the rejection region given by ’reject H0if Y <θ0’.

(f) What is the connection between the lower confidence interval found in (b) and the rejection region found in (e)?

(g) Derive an expression for type II error probability βof the rejection region in (e). It will be a function of α, θ0 and θ1 a particular value of θ that less than θ0.

(h) Suppose θ0 = 1 and one observe Y = 0.1. Calculate the P-value.

Question

Part a: A manager is evaluating production and inventory. In looking over the data, he decides that a product should be continued if it sold 23,000 over the previous year. In addition, the product is considered “popular” if it receives 50 mentions by the local press over the past year.

In selecting a product at random from the catalog, let C be the likelihood that this particular product sold 23,000 products the past year. Let P be the likelihood that the product received the 50 or more mentions by the local press.

The analyst determines that P(C) = 0.297, P(P) = 0.162, and the probability that a product has sold 8000 items, and was ‘popular’ is 0.083. What is the probability that a randomly selected product either sold the requisite 23,000 items, or that it is ‘popular’?

Part b: Where would the analyst have come up with the probaility values for P(C) andP(P)?

1) a certain drug is used to treat asthma. in a clinical trial 23 of 265 treated subjects experience headache s. the accompanyng calculator displays shows results from a test of the claim that less than 9 % of treated subjects experience headches. use the normal distribution as an aproximation to the binomial and assume a 0.05 significance level to complete parts a throug e below

a) is the test two tailed, left tailed or right tailed?
b) what is the test statistic
c) what is P value
d) what is the null hypothesis and what do u conclude about it?
e) what is the final conclusion?

Simple Linear Regression: Suppose a simple linear regression analysis provides the following results:

and n = 24. Use this information to answer the following questions.

(a) State the model equation.

ŷ = β₀ + β₁x

ŷ = β₀ + β₁x + β₂s_b1   

ŷ = β₀ + β₁x₁ + β₂x₂

ŷ = β₀ + β₁s_b1

ŷ = β₀ + β₁s_b1

x̂ = β₀ + β₁s_b1

x̂ = β₀ + β₁y

(b) Test for a linear relationship between x and y. Use a 5% level of significance.

State the hypotheses to be tested.

H₀: β₀ = 0
H_a: β₀ ≠ 0

H₀: β₃ = 0
H_a: β₃ ≠ 0    

H₀: β₂ = 0
H_a: β₂ ≠ 0

H₀: β₁ = 0
H_a: β₁ ≠ 0

H₀: β₄ = 0
H_a: β₄ ≠ 0

Interpret the hypotheses you specified above.

H₀: None of the explanatory variables are important in explaining/predicting x.
H_a: At least one explanatory variable is important in explaining/predicting x.

H₀: There is a linear relationship between x and y.
H_a: There is no linear relationship between x and y  

H₀: There is no linear relationship between x and y.
H_a: There is a linear relationship between x and y

H₀: All of the explanatory variables are important in explaining/predicting x.
H_a: None of the explanatory variables are important in explaining/predicting x.

State the decision rule.

Reject H₀ if p > 0.025.
Do not reject H₀ if p ≤ 0.025.

Reject H₀ if p < 0.05.
Do not reject H₀ if p ≥ 0.05.     

Reject H₀ if p > 0.05.
Do not reject H₀ if p ≤ 0.05.

Reject H₀ if p < 0.025.
Do not reject H₀ if p ≥ 0.025.

State the appropriate test statistic name, degrees of freedom, test statistic value, and the associated p-value (Enter your degrees of freedom as a whole number, the test statistic value to three decimal places, and the p-value to four decimal places).

t ( ?   ) = , p = ( ? )  

State your decision.

?Reject the null hypothesis: There is a linear relationship between y and x.

?Reject the null hypothesis: There is not a linear relationship between y and x.     

?Do not reject the null hypothesis: There is a linear relationship between y and x.

?Do not reject the null hypothesis: There is not a linear relationship between y and x.

(c) What would be a typical size error of prediction when you use this regression model? (Round your answer to three decimal places.)

???

Choose the correct interpretation of the typical size error of prediction you identified above by mentally inserting the value into the blanks below.

When using this model to estimate parameters, we expect to be _______ units closer to the true value, on average.

When using this model to estimate parameters, we expect to be off by _______ units, on average.     

When using this model to make predictions, we expect to be _______ units closer to the true value, on average.

When using this model to make predictions, we expect to be off by _______ units, on average.

Regardless of your conclusions above concerning the quality of the model, use the model to answer the following questions.

(d) Use the model to make a prediction when x = 5.5. (Round your answer to three decimal places.)

???

Imagine that the actual value is 19.720 when x = 5.5. Calculate the residual. (Round your answer to three decimal places.)

???

Interpret the residual you calculated immediately above by mentally inserting the ABSOLUTE VALUE of the residual into the blanks below.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an overestimate.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an overestimate.     

?When using this model to make predictions, we expect to be off by _______ units, on average.

?When using this model to make predictions, we expect to be _______ units closer to the true value, on average.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an underestimate.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an underestimate.

(e) Use the model to make a prediction when x = 6.0. (Round your answer to three decimal places.)
???

Imagine that the actual value is 26.125 when x = 6.0. Calculate the residual. (Round your answer to three decimal places.)
???

Interpret the residual you calculated immediately above by mentally inserting the ABSOLUTE VALUE of the residual into the blanks below.

?When using this model to make predictions, we expect to be off by _______ units, on average.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an underestimate.    

? When using this model to make predictions, we expect to be _______ units closer to the true value, on average.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an underestimate.

?Our prediction was _______ units higher than the actual target value when x = 5.5. Our prediction was an overestimate.

?Our prediction was _______ units lower than the actual target value when x = 5.5. Our prediction was an overestimate.

A psychologist is interested in constructing a 95% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 72 of the 835 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible.

a. With 95% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between __and __

b. If many groups of 835 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About __ percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about __ percent will not contain the true population proportion.

In 2017, the mean age at first birth among US women living in metro counties was 27.7. Would you be more likely (or equally likely) to get a sample mean of 25 years if you randomly sampled 25 women or if you randomly sampled 50 women? Explain

. A population has a mean of 150 and a standard deviation of 16. a) What are the mean and standard deviation of the sampling distribution of the mean for N=25? b) What are the mean and standard deviation of the sampling distribution of the mean for N=50?

3. Given a test that is normally distributed with a mean of 100 and a standard deviation of 15, find: a) The probability that a sample of 73 scores will have a mean greater than 107. b) The probability that the mean of sample of 13 scores will be either less than 91 or greater than 10

The first apparently reliable datings of particular rock strata were obtained from the K/Ar method (comparing the proportions of Potassium 40 and Argon 40 in the rocks) in the 1960’s, and these resulted in an estimate of 370±20 million years. In the late 1970’s a newer method gave an age of 421±8 million years. Suppose that the K/Ar method results in a belief for the age which is gaussian with mean 370 and standard deviation 20 million years. Suppose that the model for the newer method is that the observed age will be gaussian with mean the true age and standard deviation 8 million years. How are the initial beliefs revised in the light of the results of the new method?

(A) 421±8

(B) 414±7.4

(C) 423±8.5

(D) 400±15

(E) 415±7.6

Data below shows counts of males and females in a sample population and the political party they voted for.

• Create the observations matrix using the provided R syntax below
• Select the appropriate statistical test,
• Formulate the null hypothesis in your own words,
• Compute the test statistic,
• Find the critical value in Chi square table with the corresponding degrees of freedom
• Find and interpret the statistical decision at the 0.01 significance level

# Gender and political affiliation

observed <- as.table(rbind(c(762, 327, 468), c(484, 239, 477)))

dimnames(observed) <- list(gender = c("F", "M"), party = c("Democrat","Independent", "Republican"))

observed

Suppose a corrections officer in Washington wants to test the claim that the average time convicted armed burglars with no prior convictions spend time in jail is 104.7 months with a standard deviation of 21 months. He would like to determine if the true mean jail time is actually different from what is claimed, so he collects data for a random sample of 47 such cases from court files and determines that the mean jail time is 110.4 months. The officer conducts a one sample z-test for one mean.

(a) Express the null and alternative hypotheses symbolically. (Be sure to use appropriate symbols).

(b) Determine a 90% confidence interval for the true mean jail time.

(c) Based on the confidence interval, what decision should the police officer make about his hypothesis test? Explain your answer.

We are interested in estimating the proportion of graduates at a mid-sized university who found a job within one year of completing their undergraduate degree. We can do so by creating a 95% confidence interval for the true proportion p. Suppose we conduct a survey and find out that 340 of the 430 randomly sampled graduates found jobs within one year. Assume that the size of the population of graduates at this university is large enough so that all our conditions for normality are satisfied.

(a) What is the value of the point estimate for the proportion of graduates who found a job within one year? Round your answer to two decimal places.

(b) Use the point estimate in part (a) to calculate the standard error.

(c) For a 95% confidence interval, determine the margin of error. (Be sure that you show all your steps.) Then determine the lower and upper bounds of the interval.

(d) Which of the following is an appropriate way to interpret the confidence interval obtained in part (c)?

A. There is a 95% probability that the true proportion of people who found a job within one year of graduation is between the lower bound and the upper bound.

B. We are 95% confident that the true proportion of people who found a job within one year of graduation is between the lower bound and the upper bound.

C. We are 95% confident that the proportion of people in our sample that found a job within one year of graduation is the true proportion.

D. If we were to create many 95% confidence intervals in the same way with the same sample sizes, then all of them would contain the true proportion of the graduates who found a job within one year of graduation.

1. A new breed of avocados has weights that are normally distributed with mean of µ = 110 grams and population standard deviation σ = 16 grams.

(a) What is the probability that any one randomly chosen avocado would have a weight of 120 grams or above? Shade the appropriate region in the normal distribution plot. Be sure to plot and label the observed weight on the horizontal axis. Also label the center of the distribution with the appropriate value.

(b) The heaviest 15% of avocados are sold at a higher price, as jumbo avocados. What is the weight cut-off for jumbo avocados?

(c) What is the probability that a randomly selected avocado has a weight between 90 grams and 115 grams?

(d) A local market sells avocados in bags of 10. Avocados are randomly assigned to bags. What is the probability that a randomly selected bag of 10 avocados will have an average weight above 120 grams?

given a standardized normal distribution (with a mean of 0 and a standard deviation of 1) determine the following probabilities.

A. p(Z>1.03)

b. P(Z < -0.23)

C. P( -1.96 <Z < -0.23)

D. what is the value of Z if only 11.51% of all possible Z-values are larger?

2. Family Dentistry is a dental clinic owned and managed by a well-known doctor, Dr. Phillip

Williams, who is trained and licensed to practice general dentistry as well as dental surgery. Dr. Williams wishes to use the data on the number of patients treated at his clinic for the last

24 months to estimate demand for the dental care provided by his clinic. Demand for the clinic’s

dental care (Q_D) is specified to be a linear function of the price per visit paid by patients (PFD),

average income of households in the town and surrounding areas (M), and the price charged by

the competing dentistry, Family and Cosmetics Dentistry Center (PFCDC).              (3 pts each)

QD = a + bP_FD+ cM + dP_FCDC

Month             Q_D       P_FD       M         P_FCDC

1                      1894    316      66700 201

2                      1914    316      66700 201

3                      1489    345      66700 230

4                      1531    345      64745 230

5                      1625    345      64745 230

6                      1585    345      64745 230

7                      1577    345      66528 230

8                      1509    345      66528 230

9                      1496    374      66528 288

10                    982      403      66240 288

11                    1107    403      66240 288

12                    1424    374      66240 259

13                    1237    374      66988 259

14                    1256    374      66988 259

15                    1405    374      66988 259

16                    1504    374      67833 288

17                    1448    374      67833 288

18                    818      431      67833 288

19                    1286    403      68540 288

20                    105      546      68540 431

21                    158      546      68540 431

22                    986      431      69920 288

23                    1153    403      69920 288

24                    1527    368      69920 253

a. Using Excel and the data provided in the table, run a regression to estimate the linear demand function for the clinic’s service.                                                                                                           

(Copy and paste here your computer printout).

b. Write the equation of the sample regression line here.

c. Test the overall equation for statistical significance at the 5% significance level.

d. Evaluate the statistical significance of the estimated coefficients. Assume that the owner is comfortable using parameter estimates that are statistically significant at the 5% level or better.

e. How much in the total variation in (Q_D) is explained by the regression equation? How much of the total variation in (Q_D) is unexplained by the regression equation? What other explanatory variables might be added to this equation to increase its explanatory power.

f. If the clinic charges $450 per visit and the Family and Cosmetic Dental Center (FCDC) charges $300 per visit, and the average household income in the region is expected to be $74,000. What will be the estimated number of visitors to Family Dentistry?

1. A biologist is conducting experiments with scientific cross pollination of pink roses with white, red, and orange roses. The annual experiments show that the pink can produce 60% pink and 20% white and 20% red, red can produce 40% red, 50% pink and 10% orange, orange can produce 25% orange, 50% pink and 25% white, and white can produce 50% pink and 50% white.
   1. Express the biologist situation as a Markov chain.  
   2. If the biologist started the pollination with equal number of roses then what would be the distribution after many years?

1. Using the following data, conduct a linear regression analysis (by hand). In this analysis, you are testing whether SAT scores are predictive of overall college GPA. Also, write out the regression equation. How would you interpret your results?