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Answer:
(a) <em> </em><em>n</em> : 20 50 100 500
P (-200 < <em>X</em> - <em>μ </em>< 200) : 0.2886 0.4444 0.5954 0.9376
(b) The correct option is (b).
Step-by-step explanation:
Let the random variable <em>X</em> represent the amount of deductions for taxpayers with adjusted gross incomes between $30,000 and $60,000 itemized deductions on their federal income tax return.
The mean amount of deductions is, <em>μ</em> = $16,642 and standard deviation is, <em>σ</em> = $2,400.
Assuming that the random variable <em>X </em>follows a normal distribution.
(a)
Compute the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $200 of the population mean as follows:
- For a sample size of <em>n</em> = 20
- For a sample size of <em>n</em> = 50
- For a sample size of <em>n</em> = 100
- For a sample size of <em>n</em> = 500
<em> n</em> : 20 50 100 500
P (-200 < <em>X</em> - <em>μ </em>< 200) : 0.2886 0.4444 0.5954 0.9376
(b)
The law of large numbers, in probability concept, states that as we increase the sample size, the mean of the sample () approaches the whole population mean (
).
Consider the probabilities computed in part (a).
As the sample size increases from 20 to 500 the probability that the sample mean is within $200 of the population mean gets closer to 1.
So, a larger sample increases the probability that the sample mean will be within a specified distance of the population mean.
Thus, the correct option is (b).
One of the ways to graph this is to use plug in a few x-values and get an idea of the shape. Since the x values keep getting squared, there is an exponential increase on either side of the y-axis. You can see this by plugging in a few values:
When
x=0,f(x)=0
x=1,f(x)=1^2=1
x=2,f(x)=2^2=4
x=3,f(x)=3^2=9
x=4,f(x)=4^2=16
The same holds true for negative x-values to the left of the y-axis since a negative value squared is positive. For example,
x=−1,f(x)=(−1)2=1*−1=1
x=2,f(x)=(−2)2=−2*−2=4
The graph of f(x)=x^2 is called a "Parabola." It looks like this:
