Simplify. 100x(5 - 3p) 500x - 3p 5x - 300p 500x - 300p 500x

A population has a mean of 200 and a standard deviation of 50. Suppose a sample of size 100 is selected and x is used to estimat

zmey [24]

Answer:

a) 0.6426 = 64.26% probability that the sample mean will be within +/- 5 of the population mean.

b) 0.9544 = 95.44% probability that the sample mean will be within +/- 10 of the population mean.

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:

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

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.

In this question, we have that:

$\mu = 200, \sigma = 50, n = 100, s = \frac{50}{\sqrt{100}} = 5$

a. What is the probability that the sample mean will be within +/- 5 of the population mean (to 4 decimals)?

This is the pvalue of Z when X = 200 + 5 = 205 subtracted by the pvalue of Z when X = 200 - 5 = 195.

Due to the Central Limit Theorem, Z is:

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

X = 205

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

$Z = \frac{205 - 200}{5}$

$Z = 1$

$Z = 1$ has a pvalue of 0.8413.

X = 195

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

$Z = \frac{195 - 200}{5}$

$Z = -1$

$Z = -1$ has a pvalue of 0.1587.

0.8413 - 0.1587 = 0.6426

0.6426 = 64.26% probability that the sample mean will be within +/- 5 of the population mean.

b. What is the probability that the sample mean will be within +/- 10 of the population mean (to 4 decimals)?

This is the pvalue of Z when X = 210 subtracted by the pvalue of Z when X = 190.

X = 210

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

$Z = \frac{210 - 200}{5}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772.

X = 195

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

$Z = \frac{190 - 200}{5}$

$Z = -2$

$Z = -2$ has a pvalue of 0.0228.

0.9772 - 0.0228 = 0.9544

0.9544 = 95.44% probability that the sample mean will be within +/- 10 of the population mean.

7 0

2 years ago

The x-intercept has a y coordinate of 0. True False

Ghella [55]

False, its the other way around.
The y-intercept has an x coordinate of 0.
So, false.

5 0

2 years ago

What are the two solutions of 2|2y-1| + 5 = 19

Bess [88]

Answer:

y=4

y=−3

Step-by-step explanation:

8 0

2 years ago

Read 2 more answers

What is the first step in solving this system of equations?

DiKsa [7]

Step-by-step explanation:

Multiply the first equation by 7 and the second by 2

8 0

2 years ago

Brianna has 4 pink bracelets. One third of all her bracelets are pink. How many bracelets does brianna have?

aivan3 [116]

Hey there!

Let's first evaluate this problem, and the keyword we need to use. The problem says that out of the total number of bracelets Brianna has, x, 1/3 of them are pink. The reason the word of is so important throughout math is that it signifies multiplication. For example, 1/2 of 10 is 5 because 1/2 times 10 is 5.

Now, to apply that to this situation, we need to create an equation, a mathematical statement that indicated the equilibrium or equality of two values. Since we have 1/3 of all her bracelets, x is four, we can set up our equation. Here's another good keyword- is. Is signifies an equation, because it means "equal to" mathematically. For example, 5 is equal to 20/4.

We have:

1/3(x) = 4

(One-third of x, the total amount of bracelets, is four.)

We don't want that fraction there. In order to get rid of it and keep the expression equal, we multiply both sides by 3. We have,

x = 12

Hope this helps!

8 0

2 years ago

Simplify. 100x(5 - 3p) 500x - 3p 5x - 300p 500x - 300p 500x - 300px