Mathematics: Find the z-scores for which 15% of the distribution s area lies between -z and z

Find the z-scores for which 15% of the distribution, №8384619, 21.12.2022 10:12

Mathematics

CCA has stated in 2017 that 36.5 % of its students are first generation college students.Suppose you sample 15 CCA students and ask if they are first generation college students or not, counting the number of first generation students. a. Create a binomial probability distribution (table) for this situation. k P(X = k) 0 1 2 3 4 5 b. Give the mean of the binomial distribution. $ c. Give the standard deviation of this binomial distribution. d. Compute the z-score for the outcome x = 2 students are the first generation.

Step-by-step answer

P Answered by PhD

Step-by-step explanation:

The formula for binomial distribution is expressed as

P(x = r) = nCr × p^r × q^(n - r)

Where

x represent the number of successes.

p represents the probability of success.

q = (1 - p) represents the probability of failure.

n represents the number of trials or sample.

From the information given,

p = 36.5% = 36.5/100 = 0.365

q = 1 - p = 1 - 0.365

q = 0.635

n = 15

a) P(x = 0) = 15C0 × 0.365^0 × 0.635^(15 - 0) = 0.0011

P(x = 1) = 15C1 × 0.365^1 × 0.635^(15 - 1) = 0.0095

P(x = 2) = 15C2 × 0.365^2 × 0.635^(15 - 2) = 0.038

P(x = 3) = 15C3 × 0.365^3 × 0.635^(15 - 3) = 0.095

P(x = 4) = 15C4 × 0.365^4 × 0.635^(15 - 4) = 0.16

P(x = 5) = 15C5 × 0.365^5 × 0.635^(15 - 5) = 0.21

k P(X = k)

0 0.0011

1 0.0095

2 0.038

3 0.095

4 0.16

5 0.21

b) mean = np = 15 × 0.365 = 5.475

c) standard deviation = √npq

= √15 × 0.365 × 0.635

= 1.86

d) z = (x - mean)/standard deviation

x = 2

z = (2 - 5.475)/1.86 = - 1.87

Mathematics

Suppose that the national average for the math portion of the college board s sat is 510. the college board periodically rescales the test scores such that the standard deviation is approximately 100. answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores. if required, round your answers to two decimal places. if your answer is negative use minus sign . (a) what percentage of students have an sat math score greater than 610? % (b) what percentage of students have an sat math score greater

Step-by-step answer

P Answered by Specialist

a) 0.1587 b) 0.023 c) 0.1587 d) 1.15 e)-0.95

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 510

Standard Deviation, σ = 100

We are given that the distribution of SAT score is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P(score greater than 610)

P(x > 610)

$P( x 610) = P( z \displaystyle\frac{610 - 510}{100}) = P(z 1)$

$= 1 - P(z \leq 1)$

Calculation the value from standard normal z table, we have,

$P(x 610) = 1 - 0.8413 = 0.1587 = 15.87\%$

b) P(score greater than 710)

$P(x 710) = P(z \displaystyle\frac{710-510}{100}) = P(z 2)\\\\P( z 2) = 1 - P(z \leq 2)$

Calculating the value from the standard normal table we have,

$1 - 0.977 = 0.023 = 2.3\%\\P( x 710) = 2.3\%$

c)P(score between 410 and 510)

$P(410 \leq x \leq 510) = P(\displaystyle\frac{410 - 510}{100} \leq z \leq \displaystyle\frac{510-510}{100}) = P(-1 \leq z \leq 0)\\\\= P(z \leq 0) - P(z < -1)\\= 0.500 - 0.159 = 0.341 = 34.1\%$

$P(410 \leq x \leq 510) = 34.1\%$

d) x = 625

$z_{score} = \displaystyle\frac{625 - 510}{100} = \displaystyle\frac{115}{100} = 1.15$

e) x = 415

$z_{score} = \displaystyle\frac{415 - 510}{100} = \displaystyle\frac{-95}{100} = -0.95$

Mathematics

CCA has stated in 2017 that 36.5 % of its students are first generation college students.Suppose you sample 15 CCA students and ask if they are first generation college students or not, counting the number of first generation students. a. Create a binomial probability distribution (table) for this situation. k P(X = k) 0 1 2 3 4 5 b. Give the mean of the binomial distribution. $ c. Give the standard deviation of this binomial distribution. d. Compute the z-score for the outcome x = 2 students are the first generation.

Step-by-step answer

P Answered by PhD

Step-by-step explanation:

The formula for binomial distribution is expressed as

P(x = r) = nCr × p^r × q^(n - r)

Where

x represent the number of successes.

p represents the probability of success.

q = (1 - p) represents the probability of failure.

n represents the number of trials or sample.

From the information given,

p = 36.5% = 36.5/100 = 0.365

q = 1 - p = 1 - 0.365

q = 0.635

n = 15

a) P(x = 0) = 15C0 × 0.365^0 × 0.635^(15 - 0) = 0.0011

P(x = 1) = 15C1 × 0.365^1 × 0.635^(15 - 1) = 0.0095

P(x = 2) = 15C2 × 0.365^2 × 0.635^(15 - 2) = 0.038

P(x = 3) = 15C3 × 0.365^3 × 0.635^(15 - 3) = 0.095

P(x = 4) = 15C4 × 0.365^4 × 0.635^(15 - 4) = 0.16

P(x = 5) = 15C5 × 0.365^5 × 0.635^(15 - 5) = 0.21

k P(X = k)

0 0.0011

1 0.0095

2 0.038

3 0.095

4 0.16

5 0.21

b) mean = np = 15 × 0.365 = 5.475

c) standard deviation = √npq

= √15 × 0.365 × 0.635

= 1.86

d) z = (x - mean)/standard deviation

x = 2

z = (2 - 5.475)/1.86 = - 1.87

Mathematics

Suppose that the national average for the math portion of the college board s sat is 510. the college board periodically rescales the test scores such that the standard deviation is approximately 100. answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores. if required, round your answers to two decimal places. if your answer is negative use minus sign . (a) what percentage of students have an sat math score greater than 610? % (b) what percentage of students have an sat math score greater

Step-by-step answer

P Answered by Master

a) 0.1587 b) 0.023 c) 0.1587 d) 1.15 e)-0.95

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 510

Standard Deviation, σ = 100

We are given that the distribution of SAT score is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

a) P(score greater than 610)

P(x > 610)

$P( x 610) = P( z \displaystyle\frac{610 - 510}{100}) = P(z 1)$

$= 1 - P(z \leq 1)$

Calculation the value from standard normal z table, we have,

$P(x 610) = 1 - 0.8413 = 0.1587 = 15.87\%$

b) P(score greater than 710)

$P(x 710) = P(z \displaystyle\frac{710-510}{100}) = P(z 2)\\\\P( z 2) = 1 - P(z \leq 2)$

Calculating the value from the standard normal table we have,

$1 - 0.977 = 0.023 = 2.3\%\\P( x 710) = 2.3\%$

c)P(score between 410 and 510)

$P(410 \leq x \leq 510) = P(\displaystyle\frac{410 - 510}{100} \leq z \leq \displaystyle\frac{510-510}{100}) = P(-1 \leq z \leq 0)\\\\= P(z \leq 0) - P(z < -1)\\= 0.500 - 0.159 = 0.341 = 34.1\%$

$P(410 \leq x \leq 510) = 34.1\%$

d) x = 625

$z_{score} = \displaystyle\frac{625 - 510}{100} = \displaystyle\frac{115}{100} = 1.15$

e) x = 415

$z_{score} = \displaystyle\frac{415 - 510}{100} = \displaystyle\frac{-95}{100} = -0.95$

Mathematics

A person s IQ (Intelligence Quotient) is supposed to be a measure of his or her intelligence. The IQ test scores are normally distributed and are scaled to a mean of 100 and a standard deviation of 15. It is often said that a genius is someone with an IQ of 140 or more. What is the z-score for a 140 IQ? (Round your answer to one decimal place.) z = What percentage of the population scores as a genius? Round your answer as a percentage to one decimal place. %

Step-by-step answer

P Answered by PhD

1

2.7 and 99.6%

Step-by-step explanation:

In this question, we are asked to calculate the percentage of the population scores as a genius.

We proceed as follows;

z =( Mean - value)/standard deviation= (140 - 100)/15 = 2.6667 = 2.7

From normal probability table, given that z = 2.67, the probability is 0.9962

This means that the required percentage is 99.6%

Mathematics

A person s IQ (Intelligence Quotient) is supposed to be a measure of his or her intelligence. The IQ test scores are normally distributed and are scaled to a mean of 100 and a standard deviation of 15. It is often said that a genius is someone with an IQ of 140 or more. What is the z-score for a 140 IQ? (Round your answer to one decimal place.) z = What percentage of the population scores as a genius? Round your answer as a percentage to one decimal place. %

Step-by-step answer

P Answered by PhD

1

2.7 and 99.6%

Step-by-step explanation:

In this question, we are asked to calculate the percentage of the population scores as a genius.

We proceed as follows;

z =( Mean - value)/standard deviation= (140 - 100)/15 = 2.6667 = 2.7

From normal probability table, given that z = 2.67, the probability is 0.9962

This means that the required percentage is 99.6%

Mathematics

You can buy jars of the same jam in 2 sizes large jar:440g for £1.54 small jar:340g for £1.26 By considering the price per gram, work out which jar is the best value for money. Working must be shown

Step-by-step answer

P Answered by PhD

Answer: 440 grams for 1.54 is the better value
Explanation:
Take the price and divide by the number of grams
1.54 / 440 =0.0035 per gram
1.26 / 340 =0.003705882 per gram
0.0035 per gram < 0.003705882 per gram

Mathematics

An ice cream shop offers 5 different flavors of ice cream and 9 different toppings. How many choices are possible for a single serving of ice cream with one topping?

Step-by-step answer

P Answered by PhD

For 1 flavor there are 9 topping

Therefore, for 5 different flavors there will be 5*9 choices

No of choices= 5*9

=45

Mathematics

An online media service offers video and music downloads. Each download of a song costs $1. Each download of a video costs 5 times as much as the song download. You have a credit of $70. How many songs can you purchase after buying 12 videos?

Step-by-step answer

P Answered by PhD