More arrows The archer in Exercise 30 will be shooting 200...

More arrows The archer in Exercise 30 will be shooting 200 arrows in a large competition. a) What are the mean and standard deviation of the number of bull's-eyes she might get? b) Is a Normal model appropriate here? Explain. c) Use the $68-95-99.7$ Rule to describe the distribution of the number of bull's-eyes she may get. d) Would you be surprised if she made only 140 bull'seyes? Explain.

More arrows The archer in Exercise 30 will be shooting 200 arrows in a large competition.
a) What are the mean and standard deviation of the number of bull's-eyes she might get?
b) Is a Normal model appropriate here? Explain.
c) Use the $68-95-99.7$ Rule to describe the distribution of the number of bull's-eyes she may get.
d) Would you be surprised if she made only 140 bull'seyes? Explain.

Key Concepts

Statistical Reasoning in Assessing Outcomes

Determining whether an outcome, such as achieving only 140 bull’s-eyes, is surprising involves comparing the observed result to the expected value and the variability defined by the standard deviation. If the outcome lies far outside the range defined by the 68-95-99.7 rule or falls into a tail of the distribution, it may be considered statistically unusual or unlikely.

68-95-99.7 Rule

This rule, also known as the empirical rule, applies to normal distributions and states that approximately 68% of data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. It is used to approximate probabilities and assess how typical or atypical a particular outcome is in a normally distributed context.

Mean and Standard Deviation (Binomial)

For a binomial distribution with parameters n (number of trials) and p (probability of success), the mean (expected value) is calculated as n*p, and the standard deviation is the square root of n*p*(1-p). These measures summarize the central tendency and the spread of the distribution of successes, and are key in predicting outcomes such as the number of bull’s-eyes.

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant. In contexts like the arrow competition, each shot represents an independent trial with the possibility of being a bull’s-eye (success) or not (failure).

Normal Approximation to the Binomial

When the number of trials is large and both n*p and n*(1-p) are at least 10, the binomial distribution can be approximated by a normal distribution with the same mean and standard deviation. This approximation simplifies probability calculations, especially when dealing with a large number of trials, although it is important to verify that the conditions for its validity are met.

Transcript

00:01 So we are going to still use that statistic that the probability of making a bullseye is .8, but now we are increasing the number of shots or the number of arrows to 200.

00:13 And so the mean for this setting and the number made is going to be that 200 times .8, which is 160.

00:21 And the standard deviation will be that square root of 200 times .8 times .2.

00:28 So we'll take that 160 times .2 and then we will square root that answer or take it to the power of .5.

00:37 And we get this value to be 5 .657.

00:44 And i'm just going to store that in my calculator for future reference.

00:48 Now part b asks is this model appropriate, the normal model? well n times p is 160 and n times 1 minus p is going to be that 40.

01:00 So these are both greater than or equal to 10.

01:03 So yes, this distribution is approximately normal.

01:09 Now on part c, let's change some color here.

01:12 It says use it to describe the distribution of rolls.

01:16 So the distribution will be approximately normal.

01:19 And let's redo that line since that was so bad and not very straight.

01:25 Let's try again.

01:26 Yeah, a little bit better.

01:27 So that distribution is going to look like this centered at 160.

01:33 And again, with a standard deviation of about 5 .7.

01:38 And so if we go out one standard deviation in each direction, so that would be at 165 .7...

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