1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Vadim26 [7]
3 years ago
13

A selective college would like to have an entering class of 950 students. Because not all students who are offered admission acc

ept, the college admits more than 950 students. Past experience shows that about 75% of the students admitted will accept. The college decides to admit 1200 students. Assuming that students make their decisions independently, the number who accept has the B(1200, 0.75) distribution. If this number is less than 950, the college will admit students from its waiting list. (a) What are the mean and the standard deviation of the number X of students who accept? (b) Use the Normal approximation to find the probability that at least 800 students accept. (c) The college does not want more than 950 students. What is the probability that more than 950 will accept? (d) If the college decides to increase the number of admission offers to 1300, what is the probability that more than 950 will accept?
Mathematics
1 answer:
pogonyaev3 years ago
5 0

Answer:

a) The mean is 900 and the standard deviation is 15.

b) 100% probability that at least 800 students accept.

c) 0.05% probability that more than 950 will accept.

d) 94.84% probability that more than 950 will accept

Step-by-step explanation:

We use the normal approximation to the binomial to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

\sqrt{V(X)} = \sqrt{np(1-p)}

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that \mu = E(X), \sigma = \sqrt{V(X)}.

(a) What are the mean and the standard deviation of the number X of students who accept?

n = 1200, p = 0.75. So

E(X) = np = 1200*0.75 = 900

\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1200*0.75*0.25} = 15

The mean is 900 and the standard deviation is 15.

(b) Use the Normal approximation to find the probability that at least 800 students accept.

Using continuity corrections, this is P(X \geq 800 - 0.5) = P(X \geq 799.5), which is 1 subtracted by the pvalue of Z when X = 799.5. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{799.5 - 900}{15}

Z = -6.7

Z = -6.7 has a pvalue of 0.

1 - 0 = 1

100% probability that at least 800 students accept.

(c) The college does not want more than 950 students. What is the probability that more than 950 will accept?

Using continuity corrections, this is P(X \geq 950 - 0.5) = P(X \geq 949.5), which is 1 subtracted by the pvalue of Z when X = 949.5. So

Z = \frac{X - \mu}{\sigma}

Z = \frac{949.5 - 900}{15}

Z = 3.3

Z = 3.3 has a pvalue of 0.9995

1 - 0.9995 = 0.0005

0.05% probability that more than 950 will accept.

(d) If the college decides to increase the number of admission offers to 1300, what is the probability that more than 950 will accept?

Now n = 1300. So

E(X) = np = 1300*0.75 = 975

\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{1200*0.75*0.25} = 15.6

Same logic as c.

Z = \frac{X - \mu}{\sigma}

Z = \frac{949.5 - 975}{15.6}

Z = -1.63

Z = -1.63 has a pvalue of 0.0516

1 - 0.0516 = 0.9484

94.84% probability that more than 950 will accept

You might be interested in
You have 9 math coins and each coin is worth -25 points. How many points is that all together?
lesya [120]
The answer is -225.
9 times -25 is -225.
9*-25=-225
6 0
3 years ago
Read 2 more answers
The mean height of a sample of 15-year-old boys is 175 centimeters, with a standard deviation of 10. For a sample of 15-year-old
Alina [70]
The equation for standard deviation is finding the mean of the numbers, subtracting each of the initial numbers by the mean, squaring each of the results, finding the mean of these results added together, then finding out the square root of the results added together.

The mean of 175 and 165 is 170, subtract 175 and 165 by 170, and you get 5 and -5 respectively. Then, square the results and you get 25 and 25.

Add the two numbers together (25 & 25) then find the square root of it, which is <span>7.07106781187, simplified to 7.07.</span>
4 0
2 years ago
Three times the difference of ten and a number is four.
Paul [167]

Answer:

3 (10-a) = 4 i think

Step-by-step explanation:

7 0
2 years ago
Simplify each radical expression
nlexa [21]
31.
5 \sqrt{6}  - 3 \sqrt{6}  = 2 \sqrt{6}
32.
\sqrt{2} ( \sqrt{8}  +  \sqrt{6} ) \\  \sqrt{16}  +  \sqrt{12}  \\ 4 + 2 \sqrt{3}
33.
(3 \sqrt{2}  - 2 \sqrt{5} )(4 \sqrt{2}  + 2 \sqrt{5} ) \\ 12 \sqrt{4}  + 6 \sqrt{10}  - 8 \sqrt{10}  - 4 \sqrt{25}  \\ 24 - 2 \sqrt{10}  - 20 \\ 4 - 2 \sqrt{10}
pls mark me as brainliest
3 0
3 years ago
Please explain thank you
mixas84 [53]
Okjhgfdfghjjhgfdfghjhgfdfghjsdfghjkhgfdsfghjrhrthrthtj
6 0
2 years ago
Other questions:
  • What is the digit in the one place is the sum of the digit in a dozon I know it is 12 but do I put it in on place value or the 2
    13·2 answers
  • Brian pays £275.23 a year on his car insurance. The insurance company increases the price by 2.6%. How much does the insurance c
    11·1 answer
  • A farmer wants to fence a section of land for a horse pasture. Fencing costs $27 per yard. How much will it cost to fence the pa
    8·1 answer
  • Please answer question need answer asap
    5·1 answer
  • Find the number that belongs
    10·1 answer
  • After choir practice, Jamal and Bettina shared a pizza that was cut into 10 equal slices. Jamal ate 1/2 of the pizza, and Bettin
    12·1 answer
  • Tell me ASAP plz!! Btw Patsy’s Pasta house uses 3/4 of a cup of sauce for every 12 ounces of pasta...good luck
    12·1 answer
  • Write the equation of the line in fully simplified slope-intercept form.
    10·1 answer
  • Write these equations of the line in point-slope form given the slope
    10·1 answer
  • Step 1: –10 8x &lt; 6x – 4 step 2: –10 &lt; –2x – 4 step 3: –6 &lt; –2x step 4: ________ what is the final step in solving the i
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!