1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
-Dominant- [34]
3 years ago
12

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

mean 21.1 and standard deviation 5.1. What is the probability that a single student randomly chosen from all those taking the test scores 23 or higher? P(X > 0.3548) = P(z > 0.3725) = 0.3725 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? p(X >) = p(z >) = Why is it more likely that a single student would score this high instead of the sample of students?
Mathematics
1 answer:
Mumz [18]3 years ago
5 0

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.

You might be interested in
Last week DeShawn ran 31 kilometers less than Amanda. DeShawn ran 13 kilometers. How many kilometers did Amanda run?
schepotkina [342]
31+13=44 miles ran by amanda
6 0
2 years ago
Read 2 more answers
A parabola has a vertex at (-1,0) and opens down. What is the equation of the parabola?
Lapatulllka [165]

Answer:

y = - (x + 1)²

Step-by-step explanation:

Vertex form of a parabola is given by,

y = -(x - h)² + k

Where (h, k) is the ordered pair representing vertex.

(-) sign denotes that the parabola is opening downwards.

If the coordinates of the given parabola is (-1, 0)

Equation of the parabola will be,

y = - (x + 1)² + 0

y = - (x + 1)²

Therefore, equation of the parabola will be, y = - (x + 1)²

5 0
3 years ago
Stephen, Gavin and Jim share some sweets in the ratio 4:1:2. Stephen gets 30 more sweets than Jim. How many sweets are there alt
Nutka1998 [239]

Answer:

$17.00

Step-by-step explanation:

8 0
3 years ago
Ive.
allochka39001 [22]

Answer:

50 + 200h

Step-by-step explanation:

4 0
3 years ago
How to solve 8× 10 rasied to the power of 4
Sunny_sXe [5.5K]

Answer: 80,000

Step-by-step explanation: 8x(10x10x10x10)=8x10,000=80,000

6 0
3 years ago
Other questions:
  • Evaluate variable expressions
    5·1 answer
  • 3x-6y=12<br><br> Please solve for y and show you work.
    10·1 answer
  • Mr. Evans is paid $9.20 per hour for the first 40 hours he works in a week. He is paid 1.5 times that rate for each hour he work
    7·2 answers
  • How do you solve this question?
    5·1 answer
  • The circumference of the bike tire above is 84.152 inches. What is the radius of the bike tire? (Use 3.14 for .) A. 42.08 in B.
    5·1 answer
  • The value of a car decreases by 20% per year Mr sing purchases a $22,000 automobile what is the value of that car at the end of
    11·1 answer
  • (-2,6) and (1,-1) whats the slope
    11·2 answers
  • I had 3 apples. I add 2 more apples. How many apples do I have?
    6·1 answer
  • I got D for my answer and want to make sure that it’s correct
    11·2 answers
  • To purchase 13700 worth of restaurant equipment for her business Maria made a down payment of 1500 and took out a business loan
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!