What is the solution to this equation? log

The composite scores of individual students on the ACT college entrance examination in 2009 followed a normal distribution with

Mumz [18]

Answer:

35.57% probability that a single student randomly chosen from all those taking the test scores 23 or higher.

0.41% probability that a simple random sample of 50 students chosen from all those taking the test has an average score of 23 or higher.

The lower the standard deviation, the higher the z-score, which means that the higher the pvalue of X = 23, which means there is a lower probability of scoring above 23. By the Central Limit Theorem, as the sample size increases, the standard deviation decreases, which means that Z increases.

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 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.

In this problem, we have that:

$\mu = 21.1, \sigma = 5.1$

What is the probability that a single student randomly chosen from all those taking the test scores 23 or higher?

This is the pvalue of Z when X = 23.

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

$Z = \frac{23 - 21.1}{5.1}$

$Z = 0.37$

$Z = 0.37$ has a pvalue of 0.6443

1 - 0.6443 = 0.3557

35.57% probability that a single student randomly chosen from all those taking the test scores 23 or higher.

What is the probability that a simple random sample of 50 students chosen from all those taking the test has an average score of 23 or higher?

Now we use the central limit theorem, so $n = 50, s = \frac{5.1}{\sqrt{50}} = 0.72$

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

$Z = \frac{23 - 21.1}{0.72}$

$Z = 2.64$

$Z = 2.64$ has a pvalue of 0.9959

1 - 0.9959 = 0.0041

0.41% probability that a simple random sample of 50 students chosen from all those taking the test has an average score of 23 or higher.

Why is it more likely that a single student would score this high instead of the sample of students?

The lower the standard deviation, the higher the z-score, which means that the higher the pvalue of X = 23, which means there is a lower probability of scoring above 23. By the Central Limit Theorem, as the sample size increases, the standard deviation decreases, which means that Z increases.

5 0

3 years ago

Veronica tells another classmate that ΔHAT is congruent to ΔPAM because of the AAA Theorem. Is Veronica correct in what she tell

SCORPION-xisa [38]

Answer: Veronica is not correct

Explanation:

The AAA theorem, aka the AA theorem, only applies to similar triangles. We cannot use it to prove two triangles are congruent or not. We would need to know info about at least one pair of sides. So based on this diagram, we don't have enough information to know if triangle HAT is congruent to triangle PAM or not.

A similar example to this is to consider two equilateral triangles. Let's call them triangle A and triangle B. If equilateral triangle A has side lengths of 2, and has side lengths of 10, we can see that the triangles are not congruent. However, the triangles are similar because the corresponding angles equal one another. One triangle is a scaled copy of the other. This example is a counterexample as to why the AAA theorem is not a valid congruence theorem.

3 0

2 years ago

What is the simplified form of the quantity y-squared plus 7y plus 12 over the quantity y-squared plus 8y plus 15

Burka [1]

Firstly, let's factorise each equation individually - to do this, find 2 numbers that when summed add to the value of the second term, and when multiplied give the value of the third term.

7 and 12 give us 4 and 3 (4+3=7, 4*3=12) -- 8 and 15 give us 5 and 3 (5+3=8, 5*3=15)

Now we can rewrite these equations as (y+4)(y+3) and (y+5)(y+3) respectively.

Putting this in a fraction: (y+4)(y+3)/(y+5)(y+3) -- We can clearly see that there is a y+3 on both sides of the fraction, and given there are no terms outside of the brackets being multiplied, we can directly cancel.

This gives us our final answer:
(y+4)/(y+5)

8 0

3 years ago

Evaluate7.2÷0.38.(Round to the hundredth place as needed.)

polet [3.4K]

Answer:

18.95

Step-by-step explanation:

18.94736842 rounded to its nearest hundredth will be 18.95

6 0

3 years ago

Pyramid A has a triangular base where each side measures 4 units and a volume of 36 cubic units. Pyramid B has the same height,

omeli [17]