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
KiRa [710]
3 years ago
11

The time for a visitor to read health instructions on a Web site is approximately normally distributed with a mean of 10 minutes

and a standard deviation of 2 minutes. Suppose 64 visitors independently view the site. Determine the following: a. The expected value and the variance of the mean time of the visitors. b. The probability that the mean time of the visitors is within 15 seconds of 10 minutes c. The value exceeded by the mean time of the visitors with probability 0.01.
Mathematics
1 answer:
klio [65]3 years ago
6 0

Answer:

a) The mean is 10 and the variance is 0.0625.

b) 0.6826 = 68.26% probability that the mean time of the visitors is within 15 seconds of 10 minutes.

c) 10.58 minutes.

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.

Normally distributed with a mean of 10 minutes and a standard deviation of 2 minutes.

This means that \mu = 10, \sigma = 2

Suppose 64 visitors independently view the site.

This means that n = 64,  = \frac{2}{\sqrt{64}} = 0.25

a. The expected value and the variance of the mean time of the visitors.

Using the Central Limit Theorem, mean of 10 and variance of (0.25)^2 = 0.0625.

b. The probability that the mean time of the visitors is within 15 seconds of 10 minutes.

15 seconds = 15/60 = 0.25 minutes, so between 9.75 and 10.25 seconds, which is the p-value of Z when X = 10.25 subtracted by the p-value of Z when X = 9.75.

X = 10.25

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

By the Central Limit Theorem

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

Z = \frac{10.25 - 10}{0.25}

Z = 1

Z = 1 has a p-value of 0.8413.

X = 9.75

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

Z = \frac{9.75 - 10}{0.25}

Z = -1

Z = -1 has a p-value of 0.1587.

0.8413 - 0.1587 = 0.6826.

0.6826 = 68.26% probability that the mean time of the visitors is within 15 seconds of 10 minutes.

c. The value exceeded by the mean time of the visitors with probability 0.01.

The 100 - 1 = 99th percentile, which is X when Z has a p-value of 0.99, so X when Z = 2.327.

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

2.327 = \frac{X - 10}{0.25}

X - 10 = 2.327*0.25

X = 10.58

So 10.58 minutes.

You might be interested in
Leah is building a slide for her kids. if the latter is 7 feet tall and she wants the bottom of the slide to be 4 feet from the
nadezda [96]

The line needs to be 8.06 m long

Explanation

Given that the ladder is 7 feet tall and the bottom of the slide should be 4 m away from the ladder.

The ladder,slide and the ground connecting them makes the sides of a right angled triangle with ladder as the height and ground as the base.

base=4m

height=7m

hypotenuse can be determined using pythagoras law

b^{2} +h^{2} =H^{2} \\x^{2} 4^{2}+ 7^{2}= H^{2} \\16+49=H^{2} \\H=\sqrt{65} \\=8.06 m

4 0
3 years ago
Ricardo changed $600 into pounds when the exchange rate was $1 = 0.60 pounds
yulyashka [42]

600 x 0.60 = 360 pounds

360 /0.72 = 500

He got $500

8 0
2 years ago
45 Points! Exponets! Please show ALL Work
anygoal [31]
On the first one you do 2x 2x2 • 2x2x2x2x2 = 60
8 0
3 years ago
The number of blue chips is 3 more than twice the number of red chips
Elenna [48]

Answer:

red: x

blue: 2x+3

Step-by-step explanation:

let the number of red chips= x

let the number of blue chips= 2x+3

7 0
3 years ago
Read 2 more answers
Calculate the volume​
Yakvenalex [24]
V = LXHXW
V = 12X20X8 = 1920
7 0
2 years ago
Other questions:
  • Which decimal number best describes the fraction 137/6
    15·2 answers
  • Why is it useful to factor polynomial expressions for a given scenario?
    8·1 answer
  • A gym has an inclined treadmill for joggers, as shown in the figure.
    7·1 answer
  • I have no clue on how to solve this
    5·2 answers
  • In September 2011, Gallup surveyed 1,004 American adults and asked them whether they blamed Barack Obama a great deal, a moderat
    8·1 answer
  • PLEASE HELP FOR BRAINLIEST!!
    6·1 answer
  • The angle ____ is the smallest positive coterminal angle with 915 degrees
    10·1 answer
  • Sovle, Simplify<br><br> 1 1/2 divided by 2/3
    10·1 answer
  • Which one is the answer? Answer these two
    8·2 answers
  • 3). Let g(x) = x^4 - 6x^3 + 13x^2 - 24x + 36. Find all complex zeros and multiplicities, if 2i is a complexzero. Apply the conju
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!