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Chapter 6: Problem 92

In which of the following binomial distributions does the normal distribution provide a reasonable approximation? Use computer commands to generate a graph of the distribution and compare the results to the "rule of thumb." State your conclusions. a. \(\quad n=10, p=0.3\) b. \(\quad n=100, p=0.005\) c. \(\quad n=500, p=0.1\) d. \(\quad n=50, p=0.2\)

Short Answer

Expert verified
Only for \(n=500, p=0.1\) and \(n=50, p=0.2\), the binomial distribution can be approximated by the normal distribution.

Step by step solution

01

Validate each case

Verify each binomial distribution, checking if both \(np\) and \(n(1-p)\) are at least 5. This is the indication that the normal distribution would provide a reasonable approximation.
02

Calculate np and n(1-p)

Detailed explanation for each case and their respective calculations: Case a: For \(n=10, p=0.3\), \(np = 10*0.3 = 3\) and \(n(1-p) = 10*(1-0.3) = 7\). Case b: For \(n=100, p=0.005\), \(np = 100*0.005 = 0.5\) and \(n(1-p) = 100*(1-0.005) = 99.5\). Case c: For \(n=500, p=0.1\), \(np = 500*0.1 = 50\) and \(n(1-p) = 500*(1-0.1) = 450\). Case d: For \(n=50, p=0.2\), \(np = 50*0.2 = 10\) and \(n(1-p) = 50*(1-0.2) = 40\).
03

Make conclutions,

From the calculations, it can be observed that both \(np\) and \(n(1-p)\) are at least 5 in the cases 'c' and 'd'. Therefore, it is only for these cases (c and d) that normal distribution would provide a reasonable approximation for the binomial distribution. For the other cases, at least one of the factors is less than 5, which means the normal distribution won't provide a good fit in these scenarios.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution
A binomial distribution is a way to calculate the probability of a specific number of successful outcomes in a set number of trials. Imagine flipping a coin several times; you might want to know the chance of it landing heads up a certain number of times. This is a typical binomial distribution situation.

The distribution is determined by two parameters: the number of trials (\(n\)) and the probability of success on each trial (\(p\)). It's really handy for events with binary outcomes, like win/loss or yes/no.
  • Number of Trials (\(n\)): How many times an experiment is performed.
  • Probability of Success (\(p\)): The chance of getting a "success" on a single trial.
This knowledge is crucial for understanding when a normal approximation might work.
Rule of Thumb
The normal approximation to a binomial distribution is a great analytical shortcut, but there are some basic rules. The "rule of thumb" suggests using this approximation when both \(np\) and \(n(1-p)\) are at least 5.

Why 5? This guideline ensures the binomial distribution is not too skewed and has an appropriate spread for the bell shape of the normal distribution. It's a quick check to tell whether using a normal approximation is reasonable.
  • Helps decide when normal approximation fits well.
  • Ensures the distribution isn鈥檛 overly skewed.
This rule should always be the first step before proceeding with the approximation.
Probability Calculations
Calculating probabilities in a binomial distribution involves multiplying combinations with the probability of successes and failures. For instance, to calculate \(np\) (the expected number of successes), you simply multiply \(n\) by \(p\). Similarly, \(n(1-p)\) represents the expected number of failures.

For cases like:
  • Case a: \(np = 3\), \(n(1-p) = 7\).
  • Case b: \(np = 0.5\), \(n(1-p) = 99.5\).
  • Case c: \(np = 50\), \(n(1-p) = 450\).
  • Case d: \(np = 10\), \(n(1-p) = 40\).
These calculations reveal the distribution's characteristics, helping determine if they fit the "rule of thumb."
Distribution Graph Analysis
To visually understand if a normal approximation is suitable, analyzing distribution graphs is invaluable. Visual tools like graphs can show whether a distribution resembles a bell curve. Graphs provide an immediate insight into how the data is spread or skewed.
  • Normal distributions shape like a bell.
  • If a binomial distribution closely matches this, normal approximation works.
Cases like \(c\) and \(d\) offer a good visual representation since they meet the "rule of thumb" requirement. Graphs complement quantitative checks, solidifying conclusions made from calculations.

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Most popular questions from this chapter

Find the area under the standard normal curve that corresponds to the following \(z\)-values. a. Between 0 and 1.55 b. To the right of 1.55 c. To the left of 1.55 d. Between -1.55 and 1.55

Eighty-eight percent of voters would vote for a female presidential candidate if she was qualified, a poll found. The poll was conducted in February 2007 by Gallup and reported by the Pew Research Center [http://pewresearch.org]. Only \(53 \%\) of voters felt this way in \(1969 .\) Assuming \(88 \%\) is the true current proportion, what is the probability that another poll of 1125 registered voters conducted randomly will result in a. more than eight-ninths saying they would vote for a female presidential candidate, if she was qualified? b. less than \(85 \%\) saying they would vote for a female presidential candidate, if she was qualified?

As shown in Example \(6.8,\) IQ scores are considered normally distributed, with a mean of 100 and a standard deviation of 16. a. Find the probability that a randomly selected person will have an IQ score between 100 and \(120 .\) b. Find the probability that a randomly selected person will have an IQ score above \(80 .\)

"On-hold" times for callers to a local cable television company are known to be normally distributed with a standard deviation of 1.3 minutes. Find the average caller "on-hold" time if the company maintains that no more than \(10 \%\) of callers have to wait more than 6 minutes.

a. Find the standard \(z\)-score such that \(80 \%\) of the distribution is below (to the left of) this value. b. Find the standard \(z\)-score such that the area to the right of this value is 0.15. c. Find the two \(z\)-scores that bound the middle \(50 \%\) of a normal distribution.
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