Addition of signed numbers
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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 and standard deviation
, the z-score of a measure X is given by:
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 and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
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
Suppose 64 visitors independently view the site.
This means that
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
By the Central Limit Theorem
has a p-value of 0.8413.
X = 9.75
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.
So 10.58 minutes.
Given a right triangle with base, b and height, h with the angle opposide side h as θ.
The area of a triangle is given by
But,
Therefore, the area of the triangle in terms of <span>b and θ only is given by
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The area of a triangle is given by
But,
Therefore, the area of the triangle in terms of <span>b and θ only is given by
Answer:
1. The shape of the histogram created with MS Excel is approximately bell shaped and approximately evenly spread about the central (largest counts) values
The shape of the box plot created with MS Excel is approximately evenly distributed
2. The appropriate measure of central tendency for a bell shaped histogram is the mean and median, due to the approximately equal distribution about the highest frequency class
3. Part A
The 15 numbers between 1 and 20 generated by a random generator are;
1, 8, 8, 15, 4, 18, 11, 6, 17, 1, 18, 15, 10, 12, 11
The measure of center is the mean = (1+8+8+15+4+18+11+6+17+1+18+15+10+12+11)/15 = 155/15 = 31/3
The measure of spread used is the variance, s² = 32.38
= 453.333
n = 15
s² = 453./(15 - 1) = 32.38
Part B
The measure of central tendency are the mean and the median because the size of the data (n = 15), is 75% of the population (N = 20) nd therefore the data is approximately normal and can be represented by the mean and the standard deviation
Step-by-step explanation:
