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
scoundrel [369]
3 years ago
7

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

e μ. (Round your answers to four decimal places.)
Required:
a. What is the probability that the sample mean will be within +/- 5 of the population mean (to 4 decimals)?
b. What is the probability that the sample mean will be within +/- 10 of the population mean (to 4 decimals)?
Mathematics
1 answer:
zmey [24]3 years ago
7 0

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.

You might be interested in
When commercial aircraft are inspected, wing cracks are reported as nonexistent, detectable, or critical. The history of a parti
Kamila [148]

Answer:

0.226

Step-by-step explanation:

Given that:

No wing crack (x) = 70%

Detectable wing crack (y) = 25%

Critical wing crack (z) = 5%

Number of samples (n) = 5

Probability that atleast a plane has critical crack:

1 - p(no critical crack)

Using the multinomial distribution calculator :

P(5,0,0) + p(4,1,0) + p(3,2, 0) + p(2,3,0) + p(1,4,0) + p(0,5,0)

From the calculator :

P(5,0,0) + p(4,1,0) + p(3,2, 0) + p(2,3,0) + p(1,4,0) + p(0,5,0) = 0.7741

1 - 0.7741 = 0.2259

= 0.226

7 0
3 years ago
HELP ME PLEASE! Vector's map has the scaling missing.he knows from the lake to the youth center is 8 miles. on the map, they are
Naddik [55]
Hey there!

Since 2 goes into 8, you would simplify. You would get 1 inch and 4 miles. Next, set up a ratio. The ratio is 1 inch: 4 miles. So for every inch on the map, there is 4 miles.

I hope this helps!
7 0
3 years ago
Do not round your answer. Type in just the number. 0.48 is what % of 1 .6​
MakcuM [25]

Answer:

0.00768

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
What is the square root of 25<br><br>a.4<br>b.21<br>c.10<br>d.1​
photoshop1234 [79]

the square root of 25 is 5

5x5=25

5 0
3 years ago
Read 2 more answers
Which is a correct first step for solving this equation?<br> 2 + 7 = 2x + 5 – 4x
Mariulka [41]

Step-by-step explanation:

Bringing like terms on one side

2 + 7 - 5 = 2x - 4x

9 - 5 = - 2x

4 = - 2x

4/ - 2 = x

- 2 = x

4 0
3 years ago
Other questions:
  • What are the fractions between 5 and 6 with an interval of 1/3
    15·1 answer
  • Suppose the expected returns on equity of two firms, Macrosoft and Microsoft, that operate in the same business are 10.50% and 1
    12·1 answer
  • Enter the fraction 1/5 as a decimal
    5·2 answers
  • Can any one do B and B please help me I need help me please thank you
    6·1 answer
  • Graph g(x)= 1/3x^2 and f(x) =x^2
    14·1 answer
  • What is the value of X+1/9=1/3
    13·2 answers
  • 1. 3х – 4 y = 24 : (0, ) ( ,0)​
    12·1 answer
  • maria had $150 in gift certificates to use in a record store.she bought fewer than 20 recordings.each tape cost $5.95 and each c
    14·1 answer
  • SOMEONE PLS HELP ME I REALLY NEED THIS
    13·2 answers
  • Please helppp quickly
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!