The acceptance rate for a university is 42 percent. Assuming the binomial to the normal distribution. Of the next 10 applications what is the probability that they will accept: (WHEN ROUNDING DO NOT ROUND ANY CALCULATION UNTIL THE VERY END)

a) Find the mean: μ = n ⋅ p =  round to a single decimal.

b) Calculate the standard deviation. σ = n ⋅ p ⋅ q =  . Round to 4 decimals.

c) Determine the probability that exactly four applicants will be accepted.  round to 4 decimals

d) Determine the probability that between four and six applications will be accepted.  round to 4 decimals.

e) Determine the probability that at least two of the applicants will be accepted. (You may use the complement rule here).  round to 4 decimals.

f) What is the probability for the university to accept none or more than 7 applications?  Round to 4 decimals. Would it be unusual for the university to accept no or more than 7 applicants?  Yes/No.

Question 3

In a bag of M & M’s there are 80 M & Ms, with 11 red ones, 12 orange ones,
20 blue ones, 11 green ones, 18 yellow ones, and 8 brown ones. They are
mixed up so that each candy piece is equally likely to be selected if we pick
one.
a) If we select one at random, what is the probability that it is red?
b) If we select one at random, what is the probability that it is not blue?
c) If we select one at random, what is the probability that it is red or orange?
d) If we select one at random, then put it back, mix them well and select
another one, what is the probability that both the first and second ones are
blue?
e) If we select one, keep it, and then select a second one, what is the
probability that the first one is red and the second one is green?

The amount of water in a bottle is approximately normally distributed with a mean of 2.40 liters with a standard deviation of 0.045 liter. Complete parts​ (a) through​ (e) below. a).What is the probability that an individual bottle contains less than 2.36 ​liters? b). If a sample of 4 bottles is​ selected, what is the probability that the sample mean amount contained is less than 2.36 ​liters? c).If a sample of 25 bottles is​ selected, what is the probability that the sample mean amount contained is less than 2.36 ​liters? d.) Explain the difference in the results of​ (a) and​ (c). Part​ (a) refers to an individual​ bottle, which can be thought of as a sample with sample size .1875 nothing. ​Therefore, the standard error of the mean for an individual bottle is 01 nothing times the standard error of the sample in​ (c) with sample size 25. This leads to a probability in part​ (a) that is ▼ the probability in part​ (c).

You’re waiting for Caltrain. Suppose that the waiting times have a mean of 12 minutes and a standard deviation of 3 minutes. Use the Chebyshev inequality to answer each of the following questions:

a) What is the largest possible probability that you’ll end up waiting either less than 6 minutes or more than 18 minutes for the train?

b) What is the smallest possible probability that you’ll wait between 6 and 18 minutes for the train?

c) What is the smallest possible probability that you’ll wait between 3 and 21 minutes for the train?

d) What is the smallest possible probability that you’ll wait between 0 and 24 minutes for the train?

e) Based on your answer to part (d), what is the largest possible probability that you’ll need to wait more than 24 minutes for the train? Why is this answer so dramatically different from your answer to #1c above?

PROBABILITY DISTRIBUTIONS

In a traffic study of a street in Ipswich, QLD, the following information was gathered.

- Cars passed by at an average rate of 300 cars per hour.

- The speed of the cars was normally distributed, with an average speed of 58 km/h and a variance of 2 km²/h².

Based on this information, you are asked to solve the likelihoods of certain events happening. For each question clearly indicate the random variable and the distribution it follows, solve by hand and check your answer using MATLAB.

1. What is the probability that there is less than 10 seconds time difference between one car and the next?

2. What is the probability that more than 3 cars pass by in a minute?

3. The speed limit of the road is 60 km=h. What is the probability that a random car is speeding?

4. What is the probability that there are no speeding cars within a 10 minute period?

(5) Suppose x has a distribution with μ = 20 and σ = 16.

(a) If a random sample of size n = 47 is drawn, find μ_x, σ_x and P(20 ≤ x ≤ 22). (Round σ_x to two decimal places and the probability to four decimal places.)

μ_x =

σ _x =

P(20 ≤ x ≤ 22)=

(b) If a random sample of size n = 61 is drawn, find μ_x, σ_x and P(20 ≤ x ≤ 22). (Round σ_x to two decimal places and the probability to four decimal places.)

μ_x =

σ _x =

P(20 ≤ x ≤ 22)=

c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).)
The standard deviation of part (b) is (Blank)?  part (a) because of the ( Blank) ? Sample size. Therefore, the distribution about μ_x is (Blank) ?

(8) Let x be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12 hour fast. Assume that for people under 50 years old, x has a distribution that is approximately normal, with mean μ = 54 and estimated standard deviation σ = 11. A test result x < 40 is an indication of severe excess insulin, and medication is usually prescribed.

(a) What is the probability that, on a single test, x < 40? (Round your answer to four decimal places.)

(b) Suppose a doctor uses the average x for two tests taken about a week apart. What can we say about the probability distribution of x? Hint: See Theorem 6.1.

What is the probability that x < 40? (Round your answer to four decimal places.)

(c) Repeat part (b) for n = 3 tests taken a week apart. (Round your answer to four decimal places.)

(d) Repeat part (b) for n = 5 tests taken a week apart. (Round your answer to four decimal places.)

Explain what this might imply if you were a doctor or a nurse.

(9) Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a certain article, the mean of the x distribution is about $27 and the estimated standard deviation is about $9.

(a) Consider a random sample of n = 100 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?

Is it necessary to make any assumption about the x distribution? Explain your answer.

(b) What is the probability that x is between $25 and $29? (Round your answer to four decimal places.)

(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $25 and $29? (Round your answer to four decimal places.)

(d) In part (b), we used x, the average amount spent, computed for 100 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen?

(10) A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 325 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Suppose x has mean μ = 1.4% and standard deviation σ = 1.1%.

(a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 325 stocks in the fund) has a distribution that is approximately normal? Explain.

(Blank)  x is a mean of a sample of n = 325 stocks. By the(Blank)  the x distribution( Blank) approximately normal?

(b) After 9 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)(c)After 18 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.

(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen?

(e) If after 18 months the average monthly percentage return x is more than 2%, would that tend to shake your confidence in the statement that μ = 1.4%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)  P(x > 2%)

Explain.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 15 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.22 gram.

When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

z_c =

(a)

Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limitupper limitmargin of error

(b)

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

normal distribution of weightsuniform distribution of weightsσ is knownσ is unknownn is large

(c)

Interpret your results in the context of this problem.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.     There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

(d)

Which equation is used to find the sample size n for estimating μ when σ is known?

n =

n =

     n =

n =

Find the sample size necessary for an 80% confidence level with a maximal margin of error  E = 0.08 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

hummingbirds

EcoClear Products manufactures RatX, a 100% non-toxic one-of-a-kind natural humane method to kill rats and mice that is 100% safe for people, pets and all other wildlife. Quality control personnel at EcoClear Products have determined that Bags of RatX have a population mean of 16 ounces and a population standard deviation of 4 ounces.

Part 1

A random sample of 64 bags of RatX is drawn. The probability that the average weight of the bags of RatX will be more than 17.26 ounces is . Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 4 decimal places, using conventional rounding rules.

Part 2

A random sample of 64 bags of RatX is drawn. The probability that the average weight of the bags of RatX will be between 15.36 and 15.68 ounces is . Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 4 decimal places, using conventional rounding rules.

Part 3

A random sample of 64 bags of RatX is drawn. The probability that the average weight of the bags of RatX will be less than 17.21 ounces is . Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 4 decimal places, using conventional rounding rules.

Part 4

Assume that random samples of 64 bags of RatX have been repeatedly drawn from the population and the mean number of ounces in each sample is calculated. Eighty-six percent (86%) of the sample means should be above ounces. Use only the appropriate formula and/or statistical table in your textbook to answer this question. Report your answer to 2 decimal places, using conventional rounding rules.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 13 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.36 gram.

When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

zc =

(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.) lower limit

upper limit

margin of error

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

n is large

uniform distribution of weights

normal distribution of weights

σ is unknown

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

There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.

(d) Which equation is used to find the sample size n for estimating μ when σ is known?

n = zσ σ E 2

n = zσ σ E

n = zσ E σ 2

n = zσ E σ

Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.09 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 12 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.22 gram.

When finding an 80% confidence interval, what is the critical value for confidence level? (Give your answer to two decimal places.)

Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)

lower limit

upper limit

margin of error

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

uniform distribution of weights

normal distribution of weights

n is large

σ is unknown

σ is known

Interpret your results in the context of this problem.

There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.     

The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.

The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.

There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.

Which equation is used to find the sample size n for estimating μ when σ is known?

n =

n =

n =

n =

Find the sample size necessary for an 80% confidence level with a maximal margin of error  E = 0.09 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)