Mathematics, 28.06.2019 13:30 mitch729

Monthly sales are independent normal random variables with mean 100 and standard deviation 5. (a) find the probability that exactly 3 of the next 6 months have sales greater than 100. (b) find the probability that the total of the sales in the next 4 months is greater than 420

Answers: 1

Show answers

Answers

The probability that exactly 3 of the next 6 months have sales greater than 100 is 0.31 and the probability that the total of the sales in the next 4 months is greater than 420 is 0.42.

Given :

Monthly sales are independent normal random variables with a mean of 100 and a standard deviation of 5.

The probability is given by the formula:

$\rm p = P(X100) = P\left(\dfrac{X-100}{5 }\dfrac{100-100}{5}\right)=P(Z0)=\dfrac{1}{2}$

a) The probability that exactly 3 of the next 6 months have sales greater than 100 is given by:

$\rm P(Y=3)=\dfrac{6}{3}\times \left(\dfrac{1}{2}\right)^2\times \left(1-\dfrac{1}{2}\right)^{6-3}$

$\rm P(Y=3)=\dfrac{5}{16}\approx 0.31$

b) The probability that the total of the sales in the next 4 months is greater than 420 is given by:

$\rm P(W420) = P\left(\dfrac{W-400}{100 }\dfrac{420-400}{100}\right)=P(Z0.2)\approx 0.42$

For more information, refer to the link given below:

link

Answer from: kat2788

Let be the random variable for the value of goods sold in any given month. Also let be the random variable representing the number of times that sales in a given month surpass a value of 100.

follows a normal distribution with mean $\mu=100$ and standard deviation $\sigma=5$ , while would follow a binomial distribution with some success probability on n=6 trials. To find , we use the distribution of :

$p=P(X100)=P\left(\dfrac{X-100}5\dfrac{100-100}5\right)=P(Z0)=\dfrac12$

where is a random variable following the standard normal distribution.

a) The probability that exactly 3 of the next 6 months have sales greater than 100 is then

$P(Y=3)=\dbinom63\left(\dfrac12\right)^3\left(1-\dfrac12\right)^{6-3}=\dfrac{6!}{3!(6-3)!}\cdot\dfrac1{2^6}=\dfrac5{16}\approx0.31$

b) Let be the random variable representing the total value of sales over 4 months, and denote by X_i the random variable for the value of sales during month . Then W=X_1+X_2+X_3+X_4 , and we want to find P(W420) .

We know the X_i are mutually independent and identically distributed, and we (you probably should, anyway) also know that the sum of i.i.d. normally distributed random variables also follows a normal distribution, whose mean is the sum of the means, and whose standard deviation is the sum of the squares of the standard deviations, of the i.i.d. random variables:

$X_i\sim\mathcal N(100,5)\implies W\sim\mathcal N(400,100)$

Then the probability we want is

$P(W420)=P\left(\dfrac{W-400}{100}\dfrac{420-400}{100}\right)=P(Z0.2)\approx0.42$

Answer from: Quest

ab: 2 n83b8eb7ebi. 8e yen in

Answer from: Quest

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 19:30

Plz.yesterday, the snow was 2 feet deep in front of archie’s house. today, the snow depth dropped to 1.6 feet because the day is so warm. what is the percent change in the depth of the snow?

Answers: 1

Answer

Mathematics, 21.06.2019 20:10

In the diagram, the ratios of two pairs of corresponding sides are equal to prove that almn - axyz by the sas similarity theorem, it also needs to be shown that x 6 z un z zz un = 2x zləzz lezy

Answers: 1

Answer

Mathematics, 22.06.2019 01:00

How would you find the height of points on the circle? the topic is trigonometry.

Answers: 3

Answer

Mathematics, 22.06.2019 01:30

In 2000, for a certain region, 27.1% of incoming college freshman characterized their political views as liberal, 23.1% as moderate, and 49.8% as conservative. for this year, a random sample of 400 incoming college freshman yielded the frequency distribution for political views shown below. a. determine the mode of the data. b. decide whether it would be appropriate to use either the mean or the median as a measure of center. explain your answer. political view frequency liberal 118 moderate 98 conservative 184 a. choose the correct answer below. select all that apply. a. the mode is moderate. b. the mode is liberal. c. the mode is conservative. d. there is no mode.

Answers: 1

Answer

You know the right answer?

Monthly sales are independent normal random variables with mean 100 and standard deviation 5. (a) fi...

Questions