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
Viktor [21]
3 years ago
9

The overhead reach distances of adult females are normally distributed with a mean of 197.5 cm197.5 cm and a standard deviation

of 8.3 cm8.3 cm. a. Find the probability that an individual distance is greater than 210.90210.90 cm. b. Find the probability that the mean for 1515 randomly selected distances is greater than 196.00 cm.196.00 cm. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?
Mathematics
1 answer:
fiasKO [112]3 years ago
5 0

Answer:

a) 5.37% probability that an individual distance is greater than 210.9 cm

b) 75.80% probability that the mean for 15 randomly selected distances is greater than 196.00 cm.

c) Because the underlying distribution is normal. We only have to verify the sample size if the underlying population is not normal.

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 question, we have that:

\mu = 197.5, \sigma = 8.3

a. Find the probability that an individual distance is greater than 210.9 cm

This is 1 subtracted by the pvalue of Z when X = 210.9. So

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

Z = \frac{210.9 - 197.5}{8.3}

Z = 1.61

Z = 1.61 has a pvalue of 0.9463.

1 - 0.9463 = 0.0537

5.37% probability that an individual distance is greater than 210.9 cm.

b. Find the probability that the mean for 15 randomly selected distances is greater than 196.00 cm.

Now n = 15, s = \frac{8.3}{\sqrt{15}} = 2.14

This probability is 1 subtracted by the pvalue of Z when X = 196. Then

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

By the Central Limit Theorem

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

Z = \frac{196 - 197.5}{2.14}

Z = -0.7

Z = -0.7 has a pvalue of 0.2420.

1 - 0.2420 = 0.7580

75.80% probability that the mean for 15 randomly selected distances is greater than 196.00 cm.

c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

The underlying distribution(overhead reach distances of adult females) is normal, which means that the sample size requirement(being at least 30) does not apply.

You might be interested in
During summer vacation, Erin read, on average, 4 pages per night. Once she returned to school, she averaged 3 pages per night. W
alexandr1967 [171]

Answer:

25% decrease.

Step-by-step explanation:

3/4=75%

100%-75%=25% decrease.

6 0
3 years ago
I wished I’d bought the game last week for $199.00. I missed the sale, and now it’s an additional 25%. What will it cost me now
kumpel [21]
199•25%=49.75
49.75+199=248.75
248.75•7.5%=18.65625
18.65625+248.75=267.40625
therefore the answer is $248.41
4 0
3 years ago
What is the mean of this data set? {6, 11, 5, 2, 7} Enter your answer as a decimal in the box.
Elenna [48]

Answer:

The mean is 6.2.

Step-by-step explanation:

The "mean" is the same thing as the "average". Essentially, the question is asking for the average of the numbers.

So:

Add up all of the terms. [6 + 11 + 5 + 2 + 7] = 31

You find the average by dividing the sum of the terms (31) by the number of terms (5).

31/5 = 6.2

The mean is 6.2.

I hope this helped! :)

4 0
2 years ago
A pair of running shoes cost 60.00. The sales tax is 3%. How much is sales tax?
____ [38]

Answer:

$1.8

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
Help!!! Due today!!!
vagabundo [1.1K]

Answer:

answer is 5.3

Step-by-step explanation:

just easy

7 0
3 years ago
Other questions:
  • Cone A has a height of 3 meters and a radius of 2 meters. Cone B has the same radius, but the height is 6 meters. Calculate the
    7·1 answer
  • 3(x) + 1 = 10<br><br> what’s the answer to this?
    9·1 answer
  • A 10-foot board is to be cut into 3 pieces. Two of the pieces will be the same length and one piece will be 2 feet longer than t
    15·2 answers
  • - Which set of numbers contains only
    7·2 answers
  • What is the total of equal parts of 1/10
    12·2 answers
  • The gross federal debt y​ (in trillions of​ dollars) for a certain country in year x is approximated by the following​ equation,
    10·1 answer
  • Help calculus module 6 DBQ<br><br> please show work
    10·1 answer
  • Bayes' rule can be used to identify and filter spam emails and text messages. This question refers to a large collection of real
    7·1 answer
  • Help please hurry I don’t have much time!!!
    12·2 answers
  • The sum of two numbers is 90 and the greater number exceeds twice the smaller by 15. Find the smaller number.
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!