A government entity sets a Food Defect Action Level​ (FDAL) for the various foreign substances that inevitably end up in the foods we eat. The FDAL level for insect filth in peanut butter is

0.20.insect fragment​ (larvae, eggs, body​ parts, and so​ on) per gram. Suppose that a supply of peanut butter contains 0.20.insect fragment per gram. Compute the probability that the number of insect fragments in a 6​-gram sample of peanut butter is ​(a) exactly two. Interpret the results.​(b) fewer than two. Interpret the results.​(c) at least two. Interpret the results.​(d) at least one. Interpret the results.​(e) Would it be unusual for a 6​-gram sample of this supply of peanut butter to contain four or more insect​ fragments?

1. A gene can have many alleles. Why is it that any given person only carries two copies of each allele? Pick one below

2. The X-linked recessive trait of color-blindness is present in 5% of males. If a mother who is a carrier and father who is unaffected plan to have 2 children, what is the probability the children will both be male and color-blind?
Please enter a percentage number with up to three decimals

A government entity sets a Food Defect Action Level​ (FDAL) for the various foreign substances that inevitably end up in the foods we eat. The FDAL level for insect filth in peanut butter is 0.9 insect fragment​ (larvae, eggs, body​ parts, and so​ on) per gram. Suppose that a supply of peanut butter contains 0.9 insect fragment per gram. Compute the probability that the number of insect fragments in a 9​-gram sample of peanut butter is

​(a) exactly six. Interpret the results.

​(b) fewer than six. Interpret the results. ​

(c) at least six. Interpret the results.

​(d) at least one. Interpret the results.

​(e) Would it be unusual for a 9​-gram sample of this supply of peanut butter to contain eight or more insect​ fragments?

Binomial distribution. Tables must be completed in excel only.

1. A report to Health Canada indicated that an intensive education program was used in an attempt to lower the rate of lung cancer development among females. After running the program, a long-term follow-up study of 505 females was conducted, which revealed the current rate of lung cancer development is 7.5 %.

1. Find the probability that 10 or less females have lung cancer development.
2. Assuming the program has effect, find the mean and standard deviation for the number of lung cancer cases in groups of 505 females.      
3. Considering that unusual values are outside the interval μ-2σ, μ+2σ, is 20 females out of 505 females with lung cancer development unusually low or not? Show calculations.   

The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.7% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status.

Construct the probability distribution function (PDF). (Round your probabilities to five decimal places.)

Listed below are the weights of a random sample of blue M&Ms (in grams): 0.881 0.863 0.775 0.854 0.810 0.858 0.818 0.768 0.803 0.833 0.742 0.832 0.807 0.841 0.932 (a) Create a vector with these data. Find the mean, standard deviation and number of observations for these data. (b) Draw a histogram and a normal probability plot for these data. Is the assumption of normality valid for these data? (c) Test the claim that the mean weight of all blue M&Ms is greater than 0.82 grams (α = 0.05). Include the null and alternative hypotheses and your conclusion in the context of the data. (e) Create a plot that includes the sampling distribution of your statistic under the null hypothesis, the value of the statistic as a vertical line, and the P-value. R code

R-Code

3. A rival music streaming company wishes to make inference for the proportion of individuals in the United States who subscribe to Spotify. They plan to take a survey. Let S1, . . . , Sn be the yet-to-be observed survey responses from n individuals, where the event Si = 1 corresponds to the ith individual subscribing to Spotify and the event Si = 0 corresponds to the ith individual does not subscribe to Spotify (i = 1, . . . , n). Assume that S1, . . . , Sn are i.i.d. Bernoulli(π).

a) What distribution does the random variable S = sum of Si from i = 1 to n have? Compute E(S) and var(S). The formulas should involve π and n.

(b) Suppose that n = 30 and π = 0.2. Run a Monte Carlo simulation with m = 10000 replications to verify the formulas for E(S) and var(S) from the previous question. That is, simulate 10000 i.i.d. copies of S and compare the observed average of these to the true mean, and the observed (sample) variance to the true variance. Comment.

(c) Let S¯ = S(n ^−1) = (n ^−1)*sum of Si from i=1 to n. What is the mean and variance of S¯?

(d) Verify your answers to the previous question by a Monte Carlo simulation with m = 10000 replications.

(e) Is S¯ a continuous random variable? Explain.

(f) Run a Monte Carlo simulation to estimate the probability P(S¯− 1/ √ n ≤ π ≤ S¯ + 1/ √ n) when π = 0.2 and n = 10, 20, 80, 160. Hint: For every n considered, do the following m = 10000 times: generate a random variable S˜ with the same distribution as S¯ and record whether |S˜−0.2| ≤ 1/ √ n. The Monte Carlo estimate of the desired probability is the number of times this happened divided by the total number of simulations, m = 10000.

Suppose that x is a binomial random variable with n = 5, p = .66, and q = .34.

(b) For each value of x, calculate p(x). (Round final answers to 4 decimal places.)

p(0) =

p(1)=

p(2)=

p(3)=

p(4)=

p(5)

(c) Find P(x = 3). (Round final answer to 4 decimal places.)

(d) Find P(x ≤ 3). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

(e) Find P(x < 3). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

P(x < 3) = P(x <= 2)

(f) Find P(x ≥ 4). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

(g) Find P(x > 2). (Do not round intermediate calculations. Round final answer to 4 decimal places.)

(h) Use the probabilities you computed in part b to calculate the mean μ_x, the variance, σ2x , and the standard deviation, σ_x, of this binomial distribution. Show that the formulas for μ_x , σ2x, and σ_x given in this section give the same results. (Do not round intermediate calculations. Round final answers to µ_x in to 2 decimal places, σ ²_x and σ_x in to 4 decimal places.)

μ_{x    ?}

σ_{x^2 ?}

σ_{x      ?}

(i) Calculate the interval [μ_x ± 2σ_x]. Use the probabilities of part b to find the probability that x will be in this interval. Hint: When calculating probability, round up the lower interval to next whole number and round down the upper interval to previous whole number. (Round your answers to 4 decimal places. A negative sign should be used instead of parentheses.)

The interval is [ ___ , ____ ]

P( ____ <= x <= ____) = _____

3. A rival music streaming company wishes to make inference for the proportion of individuals in the United States who subscribe to Spotify. They plan to take a survey. Let S1, . . . , Sn be the yet-to-be observed survey responses from n individuals, where the event Si = 1 corresponds to the ith individual subscribing to Spotify and the event Si = 0 corresponds to the ith individual does not subscribe to Spotify (i = 1, . . . , n). Assume that S1, . . . , Sn are i.i.d. Ber(π).

(a) What distribution does the random variable S = Pn i=1 Si have? Compute E(S) and var(S). The formulas should involve π and n.

(b) Suppose that n = 30 and π = 0.2. Run a Monte Carlo simulation with m = 10000 replications to verify the formulas for E(S) and var(S) from the previous question. That is, simulate 10000 i.i.d. copies of S and compare the observed average of these to the true mean, and the observed (sample) variance to the true variance. Comment. 1

(c) Let S¯ = n ⁻¹S = n ⁻¹ Pn i=1 Si . What is the mean and variance of S¯?

(d) Verify your answers to the previous question by a Monte Carlo simulation with m = 10000 replications.

(e) Is S¯ a continuous random variable? Explain.

(f) Run a Monte Carlo simulation to estimate the probability P(S¯− 1/ √ n ≤ π ≤ S¯ + 1/ √ n) when π = 0.2 and n = 10, 20, 80, 160. Hint: For every n considered, do the following m = 10000 times: generate a random variable S˜ with the same distribution as S¯ and record whether |S˜−0.2| ≤ 1/ √ n. The Monte Carlo estimate of the desired probability is the number of times this happened divided by the total number of simulations, m = 10000.

1. Roll a pair of dice (one is red and the other is green). Let A be the event that the red die is 4 or 5. Let B be the event that the green die is 1 Let C be the event that the dice sum is 7 or 8.

1. Calculate P(A), P(B), P(C)
2. Calculate P(A|C), P(A|B)
3. Are the events A and C independent?

2. Suppose box 1 has four black marbles and two white marbles, and box 2 has two black marbles and five marbles. If you picked one marble from one of the two boxes at random, what it the probability that you picked from box 1 given that the marble you picked is black?

3. A raffle has 5000 tickets with the following prizes: 1 ticket has $2000 prize, 10 tickets have $200 prize, and 20 tickets have $50 prize and 500 tickets have a $20 prize. If to buy a ticket costs $15, and X is the random variable that measures net profit:

1. Calculate the pdf table of X
2. Calculate E(X), Var(X)

4. If a fair coin is flipped 120 times, what is the probability that:

1. The number of heads is more than 70
2. The number of heads between 50 and 70?

5. According to a study, 21.1% of 507 female college students were on a diet at the time of the study.

a) Construct a 99% confidence interval for the true proportion of all female students who were on a diet at the time of this study.

b) Explain what this interval means.

c) Is it reasonable to think that only 17% of college women are on a diet?

6. A used car dealer says that the mean price of a 2008 Honda CR-V is at least $20,500. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $19,850 and a standard deviation of $1084. Is there enough evidence to reject the dealer’s claim at α = 0.05?