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2 years ago

The ages of people at a concert are normally distributed with a mean of 24 years and a standard deviation of 3.2 years.

Mathematics

1 answer:

pishuonlain [190]2 years ago

3 0

Answer:

The age of an attendee with a z-score of -1.3 would be 19.8 years.

Step-by-step explanation:

Mean age of people at concert = μ = 24

Standard Deviation = σ = 3.2

The ages of people at concert are Normally Distributed, so we can use z-distribution to model the age of the people.

We are given that the z-score of an attendee at the concert is -1.3, we have to find the actual age of the attendee. We can convert this z-score to the actual age by using the z-score formula. The formula to calculate the z score is:

$z=\frac{x-\mu}{\sigma}$

Here, x represents the actual age of the attendee. Substituting the given values, we get:

$-1.3=\frac{x-24}{3.2} \\\\ -1.3 \times 3.2 =x-24\\\\ -4.16=x-24\\\\ -4.16+24=x\\\\ x=19.8\\$

This means, the age of an attendee with a z-score of -1.3 would be 19.8 years.

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Let X represent the amount of gasoline (gallons) purchased by a randomly selected customer at a gas station. Suppose that the me

Alexus [3.1K]

Answer:

a) 18.94% probability that the sample mean amount purchased is at least 12 gallons

b) 81.06% probability that the total amount of gasoline purchased is at most 600 gallons.

c) The approximate value of the 95th percentile for the total amount purchased by 50 randomly selected customers is 621.5 gallons.

Step-by-step explanation:

To solve this question, we use the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are 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.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For sums, we can apply the theorem, with mean $\mu$ and standard deviation $s = \sqrt{n}*\sigma$

In this problem, we have that:

$\mu = 11.5, \sigma = 4$

a. In a sample of 50 randomly selected customers, what is the approximate probability that the sample mean amount purchased is at least 12 gallons?

Here we have $n = 50, s = \frac{4}{\sqrt{50}} = 0.5657$

This probability is 1 subtracted by the pvalue of Z when X = 12.

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

By the Central Limit theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{12 - 11.5}{0.5657}$

$Z = 0.88$

$Z = 0.88$ has a pvalue of 0.8106.

1 - 0.8106 = 0.1894

18.94% probability that the sample mean amount purchased is at least 12 gallons

b. In a sample of 50 randomly selected customers, what is the approximate probability that the total amount of gasoline purchased is at most 600 gallons.

For sums, so $mu = 50*11.5 = 575, s = \sqrt{50}*4 = 28.28$

This probability is the pvalue of Z when X = 600. So

$Z = \frac{X - \mu}{s}$

$Z = \frac{600 - 575}{28.28}$

$Z = 0.88$

$Z = 0.88$ has a pvalue of 0.8106.

81.06% probability that the total amount of gasoline purchased is at most 600 gallons.

c. What is the approximate value of the 95th percentile for the total amount purchased by 50 randomly selected customers.

This is X when Z has a pvalue of 0.95. So it is X when Z = 1.645.

$Z = \frac{X - \mu}{s}$

$1.645 = \frac{X- 575}{28.28}$

$X - 575 = 28.28*1.645$

$X = 621.5$

The approximate value of the 95th percentile for the total amount purchased by 50 randomly selected customers is 621.5 gallons.

5 0

2 years ago

What is the 8th term of the binomial expansion (x + y)10?

12345 [234]

We can use the Pascal's Triangle to solve this problem. This pascal's triangle is shown in the figure below. To build the triangle, begin with the number 1 at the top, then continue placing numbers below it in a triangular pattern. In this way, each number are the numbers directly above added together. So, the expression:

$(x+y)^{10}$

Can be extended as follows, for n = 10:

$(x+y)^{10}=x^{10}+10x^{9}y+45x^{8}y^2+120x^{7}y^3+210x^{6}y^4+252x^{5}y^5+210x^{4}y^6+120x^{3}y^7+45x^{2}y^8+10xy^9+y^{10}$

So, the 8th term of the binomial expansion is:

$120x^{3}y^7$

4 0

2 years ago

The instructor’s friend also plans to rent an apartment in the same complex. Use the graph to identify the y-intercept and the s

Marta_Voda [28]

Answer:

The y-intercept is 3000. The slope is -275. Therefore, your equation in slope intercept form is y = -275x +3000.

Step-by-step explanation:

Y intercept is found where x = 0 or on the y axis, so the only place on this graph where x = 0 and there is a number on the y axis is at (0, 3000), making 3000 your y intercept. The slope is figured from two points, and taking y2-y1/x2-x1. This would give you 3000-2450/2-0 which is -550/2 which is -275. That's your slope. Slope intercept form is y = mx+b, where m is the slope and b is the y intercept. Just plug in, and you get y = -275x+3000

4 0

2 years ago

Read 2 more answers

A library contains 60,000 books. 14,000 are the subject of business. 22,000 are on healthcare and and 12,000 on psychology and l

PIT_PIT [208]

Answer:

20%

Step-by-step explanation:

5 0

2 years ago

The radius of the large fish pond is 4 times the radius of the small fish pond.

Rom4ik [11]