MATHEMATICS COLLEGE

Suppose 41%41% of American singers are Grammy award winners. If a random sample of size 860860 is selected, what is the probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%5%?

Answers

Answer 1

Answer:

Answer:

99.72% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

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

$Z = (X - \mu)/(\sigma)$

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 $s = (\sigma)/(√(n))$ .

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

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{(p(1-p))/(n)}$

In this question:

$n = 860, p = 0.41$

$\mu = 0.41, s = \sqrt{(0.41*0.59)/(860)} = 0.0168$

What is the probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%?

This is the pvalue of Z when X = 0.41 + 0.05 = 0.46 subtracted by the pvalue of Z when X = 0.41 - 0.05 = 0.36. So

X = 0.46

$Z = (X - \mu)/(\sigma)$

By the Central Limit Theorem

$Z = (X - \mu)/(s)$

$Z = (0.46 - 0.41)/(0.0168)$

$Z = 2.98$

$Z = 2.98$ has a pvalue of 0.9986

X = 0.36

$Z = (X - \mu)/(s)$

$Z = (0.36 - 0.41)/(0.0168)$

$Z = -2.98$

$Z = -2.98$ has a pvalue of 0.0014

0.9986 - 0.0014 = 0.9972

99.72% probability that the proportion of Grammy award winners will differ from the singers proportion by less than 5%.

Related Questions

What is 421906 rounded to the nearest hundred thousand
Thirty 7th graders were surveyed and asked their favorite sport. The results showed that 15 liked football, 7 liked baseball, 5 like basketball, and 3 like soccer. What generalization can not be made?Soccer is the least favorite sport. Half of the students like football. The students would prefer to play sports over going to school.None of the students like tennis.
If ST=19 and S lies at -4 , where could T be located?
Perform each of the following computations using the measured quantities below. a. 917.2 + 45.63 = ___________b. 3.423 x 10^-4 - 6.42 x 10^-3 =_________
What is the simplified form of (radical) 400x^100 ?200^10200^5020^1020^50

Write the ratio 41 to 100 as a percent

Answers

Answer:

$41\%$

Step-by-step explanation:

we have the ratio

$(41)/(100)$

To write as percent, multiply the ratio by 100

$(41)/(100)*100=41*100/100=41\%$

Suppose human weights are normally distributed with mean 175 and standard deviation 36 pounds. A helicopter is evacuating people from a building surrounded by zombies, and this helicopter can fit 9 people and with a maximum weight of 1800 pounds, i.e., an average of 200 pounds per person. If 9 people are randomly chosen to be loaded on this helicopter, what is the probability that it can safely lift off (i.e., the average weight of a person in the helicopter is less than 200)?

Answers

Answer:

$P(\bar X <200)=P(Z<(200-175)/((36)/(√(9)))=2.083)$

And using a calculator, excel or the normal standard table we have that:

$P(Z<2.083)=0.981$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:

$X \sim N(175,36)$

Where $\mu=175$ and $\sigma=36$

They select a sample size of n=9 people.The distribution for the sample mean $\bar X$ is given by:

$\bar X \sim N(\mu, (\sigma)/(√(n)))$

And we want to find this probability:

$P(\bar X <200)$

In order to the helicopter can safely lift off. We can use the z score formula given by:

$z = (\bar X -\mu)/((\sigma)/(√(n)))$

$P(\bar X <200)=P(Z<(200-175)/((36)/(√(9)))=2.083)$

And using a calculator, excel or the normal standard table we have that:

$P(Z<2.083)=0.981$

"I think of a number, multiply it by itself and then add 6 to the result”

Answers

Answer:

9x9+6 could be possible

Step-by-step explanation:

Hope this is what you were looking for, have a great day (;

If 60% of a given number is 18.0 what is 25% of the given number

Answers

Answer:

7.5

Step-by-step explanation:

60% = 0.6

18 / 0.6 = 30 (given number)

25% = 0.25

30 * 0.25 = 7.5

Best of Luck!

Answer:

2.7

Step-by-step explanation:

60/100x = 18

x = 10.8

25/100x = ?

25/100 (10.8) = 2.7

Suppose that the functions g and h are defined for all real numbers x as follows. gx = x − 3x
hx = 5x + 2
Write the expressions for (g - h)(x) and (g * h)(x) and evaluate (g + h)(−2).

Answers

Answer:

Step-by-step explanation:

Given the functions g(x) = x − 3x and h(x) = 5x + 2, we are to calculatae for the expression;

a) (g - h)(x) an (g * h)(x)

(g - h)(x) = g(x) - h(x)

(g - h)(x) = x − 3x -(5x+2)

(g-h)(x) = x-3x-5x-2

(g-h)(x) =-7x-2

b) (g * h)(x) = g(x) * h(x)

(g * h)(x) = (x − 3x)(5x+2)

(g * h)(x) = 5x²+2x-15x²-6x

(g * h)(x) = 5x²-15x²+2x-6x

(g * h)(x) = -10x²-4x

c) To get (g + h)(−2), we need to first calculate (g + h)(x) as shown;

(g + h)(x) an (g * h)(x)

(g + h)(x) = g(x) +h(x)

(g + h)(x) = x − 3x + (5x+2)

(g+h)(x) = x-3x+5x+2

(g+h)(x) =3x+2

Substituting x = -2 into the resulting function;

(g+h)(-2) = 3(-2)+2

(g+h)(-2) = -6+2

(g+h)(-2) = -4

Helllllpppppp pleasxee huryyyyy

Answers

Answer:

Bottom left

Step-by-step explanation:

Function: each x only has one y, and this is the only graph that fits the description

The answer is the bottom left one!

Other Questions

Find the value of each trigonometric ratio

You sell candy at a football game for $1.50 each. You paid $50 for 300 candy bars. what is the domain and range for your profit?

A city currently has 31,000 residents and is adding new residents steadily at the rate of 1200 per year. If the proportion of residents that remain after t years is given by S(t) = 1/(t + 1), what is the population of the city 7 years from now?

Which way is counterclockwise?A. The way the hands move on a clock. (To the right)B. The opposite way the hands move on a clock. (To the left)