For the following, Calculate the Probability and Type II errors. Assume an level of significance of α= 0.05 .

•Q1: For testing Ha: u < 62, true population standard deviation= 15, sample size = 64, and we miraculously know that our true population mean is really 60. What is the type II error? Power? Also is our Power up to par (at least 0.8).

•Q2: For testing Ha: u >65, true population standard deviation = 12, sample size = 36, and we miraculously know that our true population mean = 66. What is the type II error? Power? Also is our Power up to par (at least 0.8).

•Q3: For testing Ha: u ≠ 65, true population standard deviation = 12, sample size = 64, and we miraculously know that our true population mean= 65. What is our power? Type II error? Is our power up to par (at least 0.8).

•Helpful Tip: You can also use the youtube videos, labeled mentioned on the previous slide, for help.

Please help me with the Q1,2,3

Suppose that you have built a simple portfolio based on allocating 70% of your funds in a mix of S&P 500 index shares and the remaining 30% of your funds in a mix of utility sector shares. If we denote the monthly return for S&P 500 index by X and the monthly return for utility index by Y , then the portfolio monthly return variable will be R = 0.7X + 0.3Y.

(a) Knowing that ??= 0.298%, ??= 0.675%, ??= 4.453%, ??= 4.403% and the correlation between X and Y is corr(X; Y ) = ??? = 0.495, compute the expected monthly return on your portfolio and the variability in your portfolio's monthly return as measured by its variance and standard deviation.

(b) Recalculate the variance and standard deviation of the return on your portfolio knowing that the two variables are uncorrelated.

(c) Compute the variance and standard deviation of the return on your portfolio knowing that the two variables are perfectly positively correlated.

(d) Compute the variance and standard deviation of the return on your portfolio knowing that the two variables are perfectly negatively correlated.

(e) Suppose you do not know the exact mean returns for your variables X and Y and based on a year's worth of previous data, somebody estimated for you a 95% confidence interval for the mean return of your portfolio. The reported interval is [0.377%; 0.453%]. Based on this information is it true that P(?? ∈ [0.377%; 0.453%]) = 0.95. Explain.

A quick survey of peanut butter prices had standard deviation and mean of $0.26 and $3.68, respectively. Compute the area for a peanut butter jar costing more than $4.25.

We have data on the lean body mass and resting metabolic rate for 10 women who are subjects in a study of dieting.Lean body mass, given in kilograms, is a persons weight leaving out all fat. Metabolic rate, in calories burned per 24 hours,is the rate at which the body consumes energy.

Lean body mass in kilograms are: ( 32.3, 43.8, 33.4, 30.9, 42.2, 35.3, 31.5, 40.5, 45.9, 47.3 )
Metabolic rates are : ( 1202.5, 1274.5, 1200.6, 1149.7, 1339, 1313.7, 1235.4, 1211.7, 1345.6, 1434.6 )

What percentage of the variability in the metabolic rate can be explained by the body mass using linear regression?

The Food Marketing Institute shows that 16% of households spend more than $100 per week on groceries. Assume the population proportion is p = 0.16 and a sample of 600 households will be selected from the population. Use z-table.

1. Calculate  the standard error of the proportion of households spending more than $100 per week on groceries (to 4 decimals).
   
   
2. What is the probability that the sample proportion will be within +/- 0.02 of the population proportion (to 4 decimals)?
   
   
3. What is the probability that the sample proportion will be within +/- 0.02 of the population proportion for a sample of 1,300 households (to 4 decimals)?

Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data. x: 38 0 21 24 19 16 36 −22 −24 −8 y: 10 −2 27 25 16 15 15 −10 −7 −5 (a) Compute Σx, Σx2, Σy, Σy2. Σx Σx2 Σy Σy2 (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to two decimal places.) x y x s2 s (c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.) x y Lower Limit Upper Limit Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data. x: 38 0 21 24 19 16 36 −22 −24 −8 y: 10 −2 27 25 16 15 15 −10 −7 −5 (a) Compute Σx, Σx2, Σy, Σy2. Σx Σx2 Σy Σy2 (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to two decimal places.) x y x s2 s (c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.) x y Lower Limit Upper Limit Use the intervals to compare the two funds. 75% of the returns for the balanced fund fall within a narrower range than those of the stock fund. 75% of the returns for the stock fund fall within a narrower range than those of the balanced fund. 25% of the returns for the balanced fund fall within a narrower range than those of the stock fund. 25% of the returns for the stock fund fall within a wider range than those of the balanced fund. (d) Compute the coefficient of variation for each fund. (Round your answers to the nearest whole number.) x y CV % % Use the coefficients of variation to compare the two funds. For each unit of return, the stock fund has lower risk. For each unit of return, the balanced fund has lower risk. For each unit of return, the funds have equal risk. If s represents risks and x represents expected return, then s/x can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better? Explain. A smaller CV is better because it indicates a higher risk per unit of expected return. A smaller CV is better because it indicates a lower risk per unit of expected return

Cindy transmitted a four-letter word to Donald.  After her computer took the ASCII representation for the word and applied the Hamming(7,4) code, Bob received this:

   1100011 0011001 1011010 1111111 0010010 0111110 1101010

Unfortunately, there were a number of errors in the transmission. Fortunately, however, there was never more than one bit error in any 7-bit block, and hence the Hamming(7,4) code was able to fix them all. Type in the four-letter word (using letters of the alphabet) that Cindy originally sent, in lower case.  (Note: Assume that the data occurs in the format we described in class whereby the three parity bits follow the four data bits, i.e. d₁ d₂ d₃ d₄ p₁ p₂ p₃ . This may differ from the order found in descriptions of Hamming(7,4) found on the web.)

Suppose you have a bag with 6 blue marbles and 4 red marbles. Calculate the following probabilities:

a) If you randomly remove half of the marbles from the bag, what is the probability that 3 of them are blue and 2 are red?

b) Suppose that you sample with replacement from the bag: you take out a marble, record its color, then throw it back in. What is the probability that you need to repeat this process exactly 10 times before seeing your first red marble on your 10th selection?

c) Again sampling with replacement, what is the probability that you need to repeat the process exactly 10 times before seeing your 4 th red marble on your 10th selection?

d) Again sampling with replacement, what is the probability that if you sample 10 times, exactly 4 of your marbles are red?

The average driving distance and driving accuracy of drives that land in the for 8 golfers are recorded in the table to the right.

Player   Distance_(yards)   Accuracy_(%)
1 315.7 39.6
2 304.4 50.3
3 309.5 44.3
4   311.7 41.5
5 295.6    57.1
6 290.1 64.6
7   295.1 56.3
8 294.1 56.5

a.Write the equation of a​ straight-line model relating driving accuracy​ (y) to driving distance​ (x). Choose the correct answer below.

b. Fit the​ model, part a​, to the data using simple linear regression. Give the least squares prediction equation.

c. Interpret the estimated​ y-intercept of the line. Choose the correct answer below.

1-For each additional yard in​ distance, the accuracy is estimated to change by the value of the​ y-intercept.

2 -For each additional percentage in​accuracy, the distance is estimated to change by the value of the​y-intercept.

3- Since a drive with accuracy is outside the range of the sample the has no practical interpretation.

4- Since a drive with distance 0 yards is outside the range of the sample the has no practical interpretation.

d. Interpret the estimated slope of the line. Choose the correct answer below.

1-Since a drive with​ 0% accuracy is outside the range of the sample​ data, the slope has no practical interpretation.

2- Since a drive with distance 0 yards is outside the range of the sample the slope has no practical interpretation.

3-For each additional percentage in​accuracy, the distance is estimated to change by the value of the slope.

4-For each additional yard in​distance, the accuracy is estimated to change by the value of the slope.

e.A golfer is practicing a new swing to increase his average driving distance. If the golfer is concerned that his driving accuracy will be​ lower, which of the two​estimates, y-intercept or​ slope, will help determine if the​golfer's concern is​ valid?

The admissions office of a small college is asked to accept deposits from a number of qualified prospective freshmen so that, with probability about 0.95, the size of the freshman class will be less than or equal to 1,100. Suppose the applicants constitute a random sample from a population of applicants, 80% of whom would actually enter the freshman class if accepted. (Use the normal approximation.)

(a)

How many deposits should the admissions counselor accept? (Round your answer up to the nearest integer.)

(b)

If applicants in the number determined in part (a) are accepted, what is the probability that the freshman class size will be less than 1,045? (Round your answer to four decimal places.)

1. You have sampled 25 randomly selected students to find the mean test score. A 95% confidence interval for the mean came out to be between 85 and 92. Which of the following statements gives a valid interpretation of this interval?
1)If this procedure were to be repeated many times, 95% of the confidence intervals found would contain the true mean score.
2)If 100 samples were taken and a 95% confidence interval were found, 95 of them would be between 85 and 92
3)95% of the 25 students have a mean between 85 and 92

2. As standard deviation increases, samples size _____________ to achieve a specified level of confidence.
1) Increases 2)Decreases 3) No answer text provided.

3. In a manufacturing process a random sample of 9 bolts manufactured has a mean length of 3 inches with a standard deviation of 0.3. What is the 90% confidence interval for the true mean length of the bolt?
1) 2.8355 to 3.1645
2) 2.4420 to 3.5580
3) 2.814 to 3.1859

The probability of buying a movie ticket with a popcorn coupon is 0.597 and without a popcorn coupon is 0.403. If you buy 18 movie tickets, we want to know the probability that no more than 13 of the tickets have popcorn coupons.

In a clinical trial, 50 patients who received a new medication are randomly selected. It was found that 10 of them suffered serious side effects from this new medication. Let p denote the population proportion of patients suffered serious side effects from this new medication.

Find the minimum sample size needed to estimate the population proportion p with 90% confidence such that the estimate is accurate within 0.04 of p.

Claim: The standard deviation of pulse rates of adult males is more thanmore than 1111 bpm. For a random sample of 143143 adult​ males, the pulse rates have a standard deviation of 11.811.8 bpm. Complete parts​ (a) and​ (b) below