Chapter 11: Problem 12
Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.8, P(B)=0.4\), and \(P(A\) and \(B)=0.25.\) Find \(P(B\) if \(A)\).
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Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.8, P(B)=0.4\), and \(P(A\) and \(B)=0.25.\) Are events A and B independent?
Ask you to convert an area from one normal distribution to an equivalent area for a different normal distribution. Draw sketches of both normal distributions, find and label the endpoints, and shade the regions on both curves. The area above 13.4 for a \(N(10,2)\) distribution converted to a standard normal distribution
Ask you to convert an area from one normal distribution to an equivalent area for a different normal distribution. Draw sketches of both normal distributions, find and label the endpoints, and shade the regions on both curves. The upper \(5 \%\) for a \(N(10,2)\) distribution converted to a standard normal distribution
Randomization Slopes A randomization distribution is created to test a null hypothesis that the slope of a regression line is zero. The randomization distribution of sample slopes follows a normal distribution, centered at zero, with a standard deviation of 2.5 (a) Draw a rough sketch of this randomization distribution, including a scale for the horizontal axis. (b) Under this normal distribution, how likely are we to see a sample slope above \(3.0 ?\) (c) Find the location of the \(5 \%\) -tile in this normal distribution of sample slopes.
Benford's Law Frank Benford, a physicist working in the 1930 s, discovered an interesting fact about some sets of numbers. While you might expect the first digits of numbers such as street addresses or checkbook entries to be randomly distributed (each with probability \(1 / 9\) ), Benford showed that in many cases the distribution of leading digits is not random, but rather tends to have more ones, with decreasing frequencies as the digits get larger. If a random variable \(X\) records the first digit in a street address, Benford's law says the probability function for \(X\) is $$ P(X=k)=\log _{10}(1+1 / k) $$ (a) According to Benford's law, what is the probability that a leading digit of a street address is \(1 ?\) What is the probability for \(9 ?\) (b) Using this probability function, what proportion of street addresses begin with a digit greater than \(2 ?\)
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