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PFC Probability Distribution of The Binomial Questions

I’m working on a statistics test / quiz prep and need an explanation to help me learn.

Q1. A local charity is holding a prize raffle to raise money for their cause. The raffle is being run by a lottery, in which each participant picks a set of 3 numbers ranging from 1 to 21. The three winning numbers are selected at random, and the prize drawing is done only once. If you decide to buy a raffle ticket, what are your chances of winning?

Q2. The mean time for a 100 meter race at a college track meet is 13.2 seconds, with a standard deviation of 0.9 seconds. To win, the next sprinter needs to run the race in 12.5 seconds or less. Assuming this random variable is normally distributed, what is the probability of the sprinter running the race in a short enough time to take the lead?

Q3. A denim company sells its jeans both online and at a retail store. Assume that 80% of the company’s sales are retail, and 20% of sales are online.
a. What’s the probability that all of the next four pairs of jeans are sold online?
b. What’s the probability that three out of the next four pairs of jeans are sold online?
c. Use your answers from part (a) and (b) to derive a formula for p(x), the probability distribution of the binomial random variable x, the number of the next four pairs of jeans sold online.

Q4.A toy manufacturer needs to test its new product to make sure it doesn’t present a choking hazard to small children. It needs to estimate the fraction of toys that could potentially be defective, ρ, to be less than .01 with a 95% confidence interval. A sample fraction of defective toys from other testing, pÌ‚, is 0.04. How many toys would the manufacturer have to randomly sample to make sure they aren’t defective?

Q5. An environmental scientist is catching fish from the Great Lakes and testing them for the presence of an invasive parasite. After a sample analysis that included 175 rounds of testing, the scientist finds that the sample mean infection rate, μ, is 54.37 fish per round and the sample standard deviation, σ, is 7.07 fish per round.
a. Construct a 95% confidence interval for the mean infection rate in the total fish population. Explain what this figure means.
b. Explain how your work in part would have been different if the sample size had been only 12 rounds of testing instead of 175.


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