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Chapter 5: Problem 20

A meteorologist in Chicago recorded the number of days of rain during a 30 -day period. If the random variable \(x\) is defined as the number of days of rain, does \(x\) have a binomial distribution? If not, why not? If so, are both values of \(n\) and \(p\) known?

Short Answer

Expert verified
If so, what are the values of n and p? Answer: Yes, the random variable x follows a binomial distribution. The value of n is 30, but the probability of rain, p, is unknown as it's not provided in the given information.

Step by step solution

01

Identify the Trials

The meteorologist is recording the number of days of rain in a 30-day period. Here, each day is considered a trial. There are 30 trials in total. Step 2: Identify the outcomes
02

Verify the Outcomes

For every day in the 30-day period, the possible outcomes are 'rain' or 'no rain'. These are the only two outcomes, so this condition is met. Step 3: Check for independence
03

Check for Independence

The outcome of one day's weather does not impact the outcome of any other day's weather. This means the trials are independent. This condition is met. Step 4: Check if the probability of success is constant
04

Verify if Probability of Success is Constant

The probability of rain, \(p\), could be influenced by factors such as the time of year and climate patterns. However, on a day-to-day basis, we assume the probability of rain stays relatively constant during this 30-day period. This means the trials have a constant probability of success. #Conclusion# Since all conditions for a binomial distribution are met, \(x\) follows a binomial distribution. However, we don't know the values of \(n\) and \(p\), as we are only given the number of trials (30) and the definition of \(x\). We know that the value of \(n\) is 30, given it's a 30-day period. However, the probability of rain, \(p\), is unknown as it's not provided in the given information.

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Most popular questions from this chapter

A subject is taught to do a task in two different ways. Studies have shown that when subjected to mental strain and asked to perform the task, the subject most often reverts to the method first learned, regardless of whether it was easier or more difficult. If the probability that a subject returns to the first method learned is .8 and six subjects are tested, what is the probability that at least five of the subjects revert to their first learned method when asked to perform their task under stress?

In a food processing and packaging plant, there are, on the average, two packaging machine breakdowns per week. Assume the weekly machine breakdowns follow a Poisson distribution. a. What is the probability that there are no machine breakdowns in a given week? b. Calculate the probability that there are no more than two machine breakdowns in a given week.

Defective Computer Chips A piece of electronic equipment contains six computer chips, two of which are defective. Three computer chips are randomly chosen for inspection, and the number of defective chips is recorded. Find the probability distribution for \(x,\) the number of defective computer chips. Compare your results with the answers obtained in Exercise 4.90

A jar contains five balls: three red and two white. Two balls are randomly selected without replacement from the jar, and the number \(x\) of red balls is recorded. Explain why \(x\) is or is not a binomial random variable. (HINT: Compare the characteristics of this experiment with the characteristics of a binomial experiment given in this section.) If the experiment is binomial, give the values of \(n\) and \(p\).

\(P(x \leq k)\) in each case: a. \(n=20, p=.05, k=2\) b. \(n=15, p=.7, k=8\) c. \(n=10, p=.9, k=9\)
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