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
katrin [286]
3 years ago
5

Suppose 42% of the population has myopia. If a random sample of size 442 is selected, what is the probability that the proportio

n of persons with myopia will differ from the population proportion by less than 3%
Mathematics
1 answer:
sineoko [7]3 years ago
7 0

Answer:

0.7994 = 79.94% probability that the proportion of persons with myopia will differ from the population proportion by less than 3%.

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}}

Suppose 42% of the population has myopia.

This means that p = 0.42

Random sample of size 442 is selected

This means that n = 442

Mean and standard deviation:

\mu = p = 0.42

s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.42*0.58}{442}} = 0.0235

What is the probability that the proportion of persons with myopia will differ from the population proportion by less than 3%?

Proportion between 0.42 + 0.03 = 0.45 and 0.42 - 0.03 = 0.39, which is the p-value of Z when X = 0.45 subtracted by the p-value of Z when X = 0.39.

X = 0.45

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

By the Central Limit Theorem

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

Z = \frac{0.45 - 0.42}{0.0235}

Z = 1.28

Z = 1.28 has a p-value of 0.8997

X = 0.39

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

Z = \frac{0.39 - 0.42}{0.0235}

Z = -1.28

Z = -1.28 has a p-value of 0.1003

0.8997 - 0.1003 = 0.7994

0.7994 = 79.94% probability that the proportion of persons with myopia will differ from the population proportion by less than 3%.

You might be interested in
would Annabelle be able to pay off the loan with the lower montly payment in 5 years by setting aside $200 each month​
NISA [10]

Answer:

no

Step-by-step explanation:

3 0
2 years ago
Is this set closed or not closed under the operation ? Negative integers under multiplication
Romashka [77]

The set is not closed

Example:

-4 * -3 = 12  this is not a negative integer

5 0
3 years ago
Find the equation of a line that passes through (2,8) and is parallel to the graph of y=2x+6. Write the equation in slope-interc
Rashid [163]

Answer:

The equation of a parallel line would be y = 2x + 4

Step-by-step explanation:

In order to find the equation of this line, we first have to note that paralell lines have same slopes. Therefore, since the original line has a slope of 2, we know the new line will have a slope of 2. Now we can use that information along with the given point in point-slope form to find the equation.

y - y1 = m(x - x1)

y - 8 = 2(x - 2)

y - 8 = 2x - 4

y = 2x + 4

3 0
2 years ago
What is 13/6 as a whole number?
SIZIF [17.4K]

Answer: 2 1/6

Step-by-step explanation: 13 divided by 6

8 0
2 years ago
mark borrows $2,100 at an 8% simple intrest rate for 9 months. What is the amount he is being charged for borrowing the money
Taya2010 [7]

Answer:

$ 126

Step-by-step explanation:

9 months is  3/4  of one year

2100 * 3/4 * .08 = 126 dollars interest

6 0
2 years ago
Other questions:
  • If x=-3 and y=-2 <br><br><br> Please help me
    6·2 answers
  • George is 4 times Dorothy's age. If George is 56, how old is Dorothy?
    15·2 answers
  • Ami and Ria are playing a game where one of them would pick up a card and the other would guess the number. Ami picked up a card
    5·2 answers
  • I think the answer is 38 but i’m not sure.
    11·1 answer
  • Find the value of X and y from the figure​
    7·2 answers
  • PLZ HELP WITH THIS QUESTION ASAP. 20 POINTS!!! Brainliest to best answer!! Please answer with detail and show ALL steps to how y
    6·1 answer
  • Select the expression equivalent to (4x-6)-(7x+8)
    11·1 answer
  • If LM = MP, then M must be the midpoint of LP? True or False? Im so confused on this!
    14·1 answer
  • Which expression is equivalent to -0.25(16m + 12)?<br><br> PLEASE HELP
    10·2 answers
  • Find the surface area of the shape below. Round your answer to two decimal places.
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!