# 6.3 Estimating the binomial with the normal distribution  (Page 2/21)

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Imagine that it is known that only 10% of Australian Shepherd puppies are born with what is called "perfect symmetry" in their three colors, black, white, and copper. Perfect symmetry is defined as equal coverage on all parts of the dog when looked at in the face and measuring left and right down the centerline. A kennel would have a good reputation for breeding Australian Shepherds if they had a high percentage of dogs that met this criterion. During the past 5 years and out of the 100 dogs born to Dundee Kennels, 16 were born with this coloring characteristic.

What is the probability that, in 100 births, more than 16 would have this characteristic?

If we assume that one dog's coloring is independent of other dogs' coloring, a bit of a brave assumption, this becomes a classic binomial probability problem.

The statement of the probability requested is 1 − [ p ( X = 0) + p ( X = 1) + p ( X = 2)+ … + p ( X = 16)]. This requires us to calculate 17 binomial formulas and add them together and then subtract from one to get the right hand part of the distribution. Alternatively, we can use the normal distribution to get an acceptable answer and in much less time.

First, we need to check if the binomial distribution is symmetrical enough to use the normal distribution. We know that the binomial for this problem is skewed because the probability of success, 0.1, is not the same as the probability of failure, 0.9. Nevertheless, both $\mathrm{np}=10$ and $\mathrm{np}\left(1-p\right)=90$ are larger than 5, the cutoff for using the normal distribution to estimate the binomial.

[link] below shows the binomial distribution and marks the area we wish to know. The mean of the binomial, 10, is also marked, and the standard deviation is written on the side of the graph: σ = $\sqrt{npq}$ = 3. The area under the distribution from zero to 16 is the probability requested, and has been shaded in. Below the binomial distribution is a normal distribution to be used to estimate this probability. That probability has also been shaded.

Standardizing from the binomial to the normal distribution as done in the past shows where we are asking for the probability from 16 to positive infinity, or 100 in this case. We need to calculate the number of standard deviations 16 is away from the mean: 10.

$Z=\frac{x-\mu }{\sigma }=\frac{16-10}{3}=2$

We are asking for the probability beyond two standard deviations, a very unlikely event. We look up two standard deviations in the standard normal table and find the area from zero to two standard deviations is 0.4772. We are interested in the tail, however, so we subtract 0.4772 from 0.5 and thus find the area in the tail. Our conclusion is the probability of a kennel having 16 dogs with "perfect symmetry" is 0.0228. Dundee Kennels has an extraordinary record in this regard.

Mathematically, we write this as:

$1-\left[p\left(X=0\right)+p\left(X=1\right)+p\left(X=2\right)+\dots +p\left(X=16\right)\right]=p\left(X>16\right)=p\left(Z>2\right)=0.0228$

## Chapter review

The normal distribution, which is continuous, is the most important of all the probability distributions. Its graph is bell-shaped. This bell-shaped curve is used in almost all disciplines. Since it is a continuous distribution, the total area under the curve is one. The parameters of the normal are the mean µ and the standard deviation σ . A special normal distribution, called the standard normal distribution is the distribution of z -scores. Its mean is zero, and its standard deviation is one.

## Formula review

Normal Distribution: X ~ N ( µ , σ ) where µ is the mean and σ is the standard deviation.

Standard Normal Distribution: Z ~ N (0, 1).

How would you represent the area to the left of one in a probability statement?

P ( x <1)

What is the area to the right of one?

Is P ( x <1) equal to P ( x ≤ 1)? Why?

Yes, because they are the same in a continuous distribution: P ( x = 1) = 0

How would you represent the area to the left of three in a probability statement?

What is the area to the right of three?

1 – P ( x <3) or P ( x >3)

If the area to the left of x in a normal distribution is 0.123, what is the area to the right of x ?

If the area to the right of x in a normal distribution is 0.543, what is the area to the left of x ?

1 – 0.543 = 0.457

Use the following information to answer the next four exercises:

X ~ N (54, 8)

Find the probability that x >56.

Find the probability that x <30.

0.0013

X ~ N (6, 2)

Find the probability that x is between three and nine.

X ~ N (–3, 4)

Find the probability that x is between one and four.

0.1186

X ~ N (4, 5)

Find the maximum of x in the bottom quartile.

Use the following information to answer the next three exercise: The life of Sunshine CD players is normally distributed with a mean of 4.1 years and a standard deviation of 1.3 years. A CD player is guaranteed for three years. We are interested in the length of time a CD player lasts. Find the probability that a CD player will break down during the guarantee period.

1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
2. P (0< x <____________) = ___________ (Use zero for the minimum value of x .)
1. Check student’s solution.
2. 3, 0.1979

Find the probability that a CD player will last between 2.8 and six years.

1. Sketch the situation. Label and scale the axes. Shade the region corresponding to the probability.
2. P (__________< x <__________) = __________

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