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It has been estimated that 17.0 % of mutual fund shareholders are retired persons. Assuming a simple random sample of 400 shareh

kkurt [141]

Answer:

a) 0.0548 = 5.48% probability that at least 20 % of those in the sample will be retired.

b) 0.8388 = 83.88% probability that between 15 % and 21 % of those in the sample will be retired.

Step-by-step explanation:

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

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes 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 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 $s = \sqrt{\frac{p(1-p)}{n}}$

17.0 % of mutual fund shareholders are retired persons.

This means that $p = 0.17$

Assuming a simple random sample of 400 shareholders has been selected

This means that $n = 400$

Mean and standard deviation:

$\mu = p = 0.17$

$s = \sqrt{\frac{0.17*0.83}{400}} = 0.0188$

a. What is the probability that at least 20 % of those in the sample will be retired?

This is the 1 p-value of Z when X = 0.2. So

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

By the Central Limit Theorem

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

$Z = \frac{0.2 - 0.17}{0.0188}$

$Z = 1.6$

$Z = 1.6$ has a p-value of 0.9452

1 - 0.9452 = 0.0548

0.0548 = 5.48% probability that at least 20 % of those in the sample will be retired.

b. What is the probability that between 15 % and 21 % of those in the sample will be retired?

This is the p-value of Z when X = 0.21 subtracted by the p-value of Z when X = 0.15.

X = 0.21

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

$Z = \frac{0.21 - 0.17}{0.0188}$

$Z = 2.13$

$Z = 2.13$ has a p-value of 0.9834

X = 0.15

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

$Z = \frac{0.15 - 0.17}{0.0188}$

$Z = -1.06$

$Z = -1.06$ has a p-value of 0.1446

0.9834 - 0.1446 = 0.8388

0.8388 = 83.88% probability that between 15 % and 21 % of those in the sample will be retired.

3 0

3 years ago

Mr. Abernathy purchased a selection of wrenches for his shop. The total cost was $78. He bought the same number of $1.50 and $2.

Over [174]

78 = 1.50x + 2.50x + 4(x/2)) + 3((x/2) + 1))
78 = 4x + 4(x/2) + 3((x/2)+1)
78 = 4x + 2x + (3(x/2) + 3)
78 = 6x + 1.5x + 3
75 = 7.5x
10 = x
Mr. Abernathy bought 10 wrenches at $1.50, 10 wrenches at $2.50, 5 wrenches at $4, and 6 wrenches at $3.

6 0

3 years ago

What is the length of z? (Round to the nearest tenth)

navik [9.2K]

Answer:

Step-by-step explanation:

$9^{2} + 6^{2} = c^{2}$

81 + 36 = $c^{2}$

117 = $c^{2}$

c = $\sqrt{117}$

c = 10.8

c = z

4 0

3 years ago

You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy som

dmitriy555 [2]

Answer:

When you have two sets of data, one that represents an independent variable (X) and another that represents a dependent variable (Y) (it is a response of the first variable). A diagram called scatterplot can be generated to present the ordered pairs (X, Y) in the Cartesian plane to see how the two variables are related.

In this case the independent variable (X) is the age variable of the calf and the dependent variable (Y) is the weight of the calf.

The diagram can be seen in the attached file.

Step-by-step explanation:

3 0

3 years ago

The travelers are adding a new room to their house. the room will be a cube with volume 3375 ft.³.they are going to put in hardw

Talja [164]

Answer:

$ 2250

Step-by-step explanation:

3375 = s³

s = ∛3375

s = 15 ft.

floor area = s² = (15 * 15)

floor area = 225 ft²

the contractor charges $10 per ft².

therefore,

the hardwood cost = 225 x $10 = $2250

7 0

3 years ago

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