Normal Distribution Notes and Workbook

7/22/2019 Normal Distribution Notes and Workbook

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NORMAL DISTRIBUTION NOTES

A random variable can be discrete or continuous. A continuous random variable is a variable which can take on

any value over a given interval. Continuous variables are measured, not counted. Items such as height, weight

and time are continuous and can take on fractional values. Eg, a basketball player may be 6.8432 ft tall.

There are many continuous probability distributions, such as, uniform distribution, normal distribution, the t

distribution, the chi-square distribution, exponential distribution, and F distribution.

We will concentrate on the normal distribution only.

Normal Distribution or Normal Curve

The Normal distribution is probably one of the most important and

widely used continuous distribution. It is known as a normal

random variable, and its probability distribution is called a normal

distribution. The following are the characteristics of the normal

distribution:

Characteristics of the Normal Distribution:

1. It is bell shaped and is symmetrical about its mean.

2. It is asymptotic to the axis, i.e., it extends indefinitely in either direction from the mean.

3. It is a continuous distribution.

4. It is a family of curves, i.e., every unique pair of mean and standard deviation defines a different normal

distribution. Thus, the normal distribution is completely described by two parameters: mean and standard

deviation. See the following figure.

5. Total area under the curve sums to 1, i.e., the area of the distribution on each side of the mean is 0.5.6. It is unimodal, i.e., values mound up only in the center of the curve.

7. The probability that a random variable will have a value between any two points is equal to the area under the

curve between those points.

Recall the Empirical Rule

Notation: xN(, 2)

The parameters of the normal distribution are the mean, , and the standard deviation, .
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Area under the curve

Probability is given by the area under the curve. Therefore, the probability that the variable takes a single value

equals zero. Calculus is normally used to find the area under a curve which is not a part of this course. We will

use a table to find the required area.

The Standard Normal Variable, z.

A standard z table can be used to find probabilities for any normal curve problem that has been converted tozscores. For the table, refer to the formula sheet.

Thezdistribution is a normal distribution with a mean of 0 and a standard deviation of 1.

zN(0, 12)

It is necessary to standardize any other variable in order to use the standard normal tables.

x

z

If you have a z value and need to convert to anx value

zx

The fol lowing steps are helpful l when working with the normal curve problems:

1. Graph the normal distribution, and shade the area related to the probability you want to find.

2. Convert the boundaries of the shaded area from x values to the standard normal random variablezvalues

using thezformula above.

3. Use the standardztable to find the probabilities or the areas related to the z values in step 2.
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WORKSHEETNORMAL DISTRIBUTION

FI NDING AREAS UNDER THE STANDARD NORMAL CURVE

For each of the following probabilities, sketch a graph showing the area under the standard normal curve, and

use the normal tables to find the missing value.

(a)P(z 1.95) = (b)P( 0z 1.95) =

(c)P(z 2.5) = (d)P(z 0.05)

(e)P( 1.52 z 2.0) = (f) P( 2.5 z 2.5) =

(g)The value ofa where P(z a) = 0.0495 (h)The value ofb where P(zb) = 0.25


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TRUE-FALSE QUESTIONS

1. There is only one standard normal distribution. _________

2. There is an infinite number of normal distributions. __________

3. One of the characteristics of the normal distribution is that it is skewed. _________

4. One half of the unit normal distribution resembles perfectly the other half. _________

5. The unit normal distribution is unrelated to statistical inference. ___________6. The asymtotic property of the unit normal distribution means that the ends of the curve extend in

opposite directions toward the baseline of the horizontal axis and never touch it. _________

7. Standard scores are also known aszscores. _________

8. Zscores from one distribution can usually be compared withZscores from another distribution,

provided both distributions are normal in form. __________

9. The amount of curve area cut off by aZ= -1.00 is the same as the amount of curve area cut off by a

Z= +1.00, provided the curve is normal. _________

10. The mean, mode, and median occur at different points on the unit normal distribution.

11. A badly skewed distribution is a normal distribution by definition. ________

12. The normal distribution has a probability function. _________

13. A curve whose left tail tapers off more than the right tail is negatively skewed. _______

14. The mean is larger than the median in a positively skewed distribution. _________

15. The bell-shaped curve is also known as a normal distribution. __________

MULTIPLE CHOICE

1. A continuous random variable is a random variable that can:a. assume only countable values b. assume any value in one or more intervals

c. have no random sample d. assume no continuous random frequency

2.The relative frequency density for a class is obtained by dividing the:a. frequency of that class by the class width

b. relative frequency of that class by the total frequency

c. relative frequency of that class by the class width

d. frequency of that class by the relative frequency

3.For a continuous random variablex, the probability thatx assumes a value in an interval is:

a. in the range zero to 1 b. greater than 1c. less than zero d. greater than 2

4. For a continuous random variablex, the total area under the probability distribution curve of x is always:a. less than 1 b. greater than 1 c. equal to 1 d. between zero and 1

5. The normal probability distribution is applied to:a. a discrete random variable b. a continuous random variable

c. any random variable d. a subjective random variable


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6.Which of the following is nota characteristic of the normal distribution?a. The total area under the curve is 1.0

b. The curve is symmetric about the mean

c. The value of the mean is always greater than the value of the standard deviation

d. The two tails of the curve extend indefinitely

7. For a normal distribution, the zvalue for an x value that is to the left of the mean is always:

a. equal to zero b. negative c. less than 1 d. positive

8. The tails of a normal distribution curve:

a. meet the horizontal axis at z = 3.0 b. never meet or cross the horizontal axisc. cross the horizontal axis at z = 4.0 d. are nonsymmetric

13. The parameters of the normal distribution are:

a. and b.,x, and

c. , , andz d. , ,z, andx

14. Convertingzscores into raw scores means that we must know the:

a. original mean and standard deviation. b. original variance and average deviation.c. original range and standard deviation. d. original mean and variance.

15. The normal distribution has:

a. one tail b. two tails. c. three tails. d. four tails.

16. For the standard normal distribution, the area to the left ofz=l.35 is:

a. .4115 b. .0885 c. .9115 d. .5855

17. For the standard normal distribution, the area between z= 0 and z= 1.97 is:

a. .4713 b. .4761 c. .4745 d. .4756

18. For the standard normal distribution, the area between z= 0 and z=2.13 is:

a. .4834 b. .4821 c. .4893 d. .4838

19. For the standard normal distribution, the area to the right ofz=.78 is:

a. .2177 b. .7177 c. .7823 d. .2823

20. Let x have a normal distribution with a mean of 40 and a standard deviation of 5. The zvalue forx = 52 is:

a. 2.00 b. 2.40 c.1.75 d. .80

21. If the area under the standard normal curve from 0 to zis .3508 and zis positive, then zis:

a. 1.00 b. 1.40 c. 1.04 d. 2.04

22. If the area under the standard normal curve from 0 to zis .4850 and zis negative, then zis:

a.2.17 b.2.10 c.3.03 d.1.21


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WORD PROBLEM S

1. Suppose the random variable, X, is defined as follows: X ~ N (45, 52).

(a) Identify the probability distribution and name the parameters of this distribution whose

Values are given in the brackets. [3]

(b) If P(X < x) = 0.1587, find the value of x. [5]

2. Suppose the random variable, X, is defined as follows: X ~ N (16, 52).

(i).Identify the probability distribution and name the parameters of this distribution whose values aregiven in the brackets. [3]

(ii).Find the value of P(X > 21). [3]3. A random variable has the following probability distribution:X ~ N (30, 4

2). Determine

(i). P(X > 26) [4](ii). P(28.5 < X < 33.6) [4](iii) The value of k such that P(X > k) = 0.0020. [6]

4. Suppose the length of life of Longlife batteries is normally distributed with = 36 months and = 4 monthsThe company guarantees the battery to last 30 months, if not they will replace the battery free of charge.

(a) What percentage of the batteries will the company have to replace? [3]

(b) What percentage of the batteries will last between 30 months and 40 months? [3]

(c) What percentage of the batteries will last 36 months or more? [1]

(d) How long will 60% of the batteries last? [4]

5. (a) A company produces fluorescent lightbulbs whose lifetimes are normally distributed with a mean of 100

hours and a standard deviation of 46 hours.

(i) Find the probability that a randomly selected lightbulb will last at least 40 hours. [4]

(ii) Find the percentage of lightbulbs that will less than 40 hours. [2]

(iii) It is found that 26% of the lightbulbs last less than thours. Find the value oft. [3]

6. Idaho Natural Produce Corporation ships potatoes to its distributors in bags whose weights are normally

distributed with a mean weight of 50kg and standard deviation of 0.5kg.

If a bag of potatoes is selected at random from a shipment, what is the probability that it weighs

(a) less than 49kg [3]

(b) more than 51 kg [3]

(c) between 49 and 51 kg [3]

(d) If 2500 bags of potatoes are shipped to a particular distributor, approximately how many of these bags will

between 49 and 51 kg? [2]

(e) 9% of the bags of potatoes shipped to this distributor were below a certain weight, w. Find this weight. [5]

Normal Distribution Notes and Workbook

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