The U.S. 2010 federal income tax for a single person with taxable income x dollars is f(x) dollars, where T is the function defi
ned by federal law as follows. A. What is the the 2010 federal income tax for a single person whose taxable income that year was 4000 dollars? Tax = ________ dollars B. What is the 2010 federal income tax for a single person whose taxable income that year was 170000 dollars? Tax = ________ dollars C. What is the taxable income for a single person whose 2010 federal income tax was 105000 dollars? Taxable income = _______ dollars
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Answer:
(a) The probability that a randomly selected Time interval between irruption is longer than 84 minutes is 0.3264.
(b) The probability that a random sample of 13 time intervals between irruption has a mean longer than 84 minutes is 0.0526.
(c) The probability that a random sample of 20 time intervals between irruption has a mean longer than 84 minutes is 0.0222.
(d) The probability decreases because the variability in the sample mean decreases as we increase the sample size
(e) The population mean may be larger than 75 minutes between irruption.
Step-by-step explanation:
We are given that a geyser has a mean time between irruption of 75 minutes. Also, the interval of time between the eruption is normally distributed with a standard deviation of 20 minutes.
(a) Let X = <u><em>the interval of time between the eruption</em></u>
So, X ~ Normal()
The z-score probability distribution for the normal distribution is given by;
Z = ~ N(0,1)
where, = population mean time between irruption = 75 minutes
= standard deviation = 20 minutes
Now, the probability that a randomly selected Time interval between irruption is longer than 84 minutes is given by = P(X > 84 min)
P(X > 84 min) = P( >
) = P(Z > 0.45) = 1 - P(Z
0.45)
= 1 - 0.6736 = <u>0.3264</u>
The above probability is calculated by looking at the value of x = 0.45 in the z table which has an area of 0.6736.
(b) Let = <u><em>sample time intervals between the eruption</em></u>
The z-score probability distribution for the sample mean is given by;
Z = ~ N(0,1)
where, = population mean time between irruption = 75 minutes
= standard deviation = 20 minutes
n = sample of time intervals = 13
Now, the probability that a random sample of 13 time intervals between irruption has a mean longer than 84 minutes is given by = P( > 84 min)
P( > 84 min) = P(
>
) = P(Z > 1.62) = 1 - P(Z
1.62)
= 1 - 0.9474 = <u>0.0526</u>
The above probability is calculated by looking at the value of x = 1.62 in the z table which has an area of 0.9474.
(c) Let = <u><em>sample time intervals between the eruption</em></u>
The z-score probability distribution for the sample mean is given by;
Z = ~ N(0,1)
where, = population mean time between irruption = 75 minutes
= standard deviation = 20 minutes
n = sample of time intervals = 20
Now, the probability that a random sample of 20 time intervals between irruption has a mean longer than 84 minutes is given by = P( > 84 min)
P( > 84 min) = P(
>
) = P(Z > 2.01) = 1 - P(Z
2.01)
= 1 - 0.9778 = <u>0.0222</u>
The above probability is calculated by looking at the value of x = 2.01 in the z table which has an area of 0.9778.
(d) When increasing the sample size, the probability decreases because the variability in the sample mean decreases as we increase the sample size which we can clearly see in part (b) and (c) of the question.
(e) Since it is clear that the probability that a random sample of 20 time intervals between irruption has a mean longer than 84 minutes is very slow(less than 5%0 which means that this is an unusual event. So, we can conclude that the population mean may be larger than 75 minutes between irruption.