Chapter 7
Short Description
Download Chapter 7...
Description
Chapter 7 Normal Curves and Sampling Distributions Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze
The Normal Distribution • A continuous distribution used for modeling many natural phenomena. • Sometimes called the Gaussian Distribution, after Carl Gauss. • The defining features of a Normal Distribution are the mean, µ, and the standard deviation, σ.
Copyright © Cengage Learning. All rights reserved.
7|2
The Normal Curve
Copyright © Cengage Learning. All rights reserved.
7|3
Features of the Normal Curve • Smooth line and symmetric around µ. • Highest point directly above µ. • The curve never touches the horizontal axis in either direction. • As σ increases, the curve spreads out. • As σ decreases, the curve becomes more peaked around µ. • Inflection points at µ ± σ.
Copyright © Cengage Learning. All rights reserved.
7|4
Two Normal Curves Both curves have the same mean, µ = 6. Curve A has a standard deviation of σ = 1. Curve B has a standard deviation of σ = 3.
Copyright © Cengage Learning. All rights reserved.
7|5
Normal Probability • The area under any normal curve will always be 1. • The portion of the area under the curve within a given interval represents the probability that a measurement will lie in that interval.
Copyright © Cengage Learning. All rights reserved.
7|6
The Empirical Rule
Copyright © Cengage Learning. All rights reserved.
7|7
The Empirical Rule
Copyright © Cengage Learning. All rights reserved.
7|8
The Empirical Rule The masses of the adult ostriches at the Vilas Zoo are normally distributed with a mean of 124 kg and a standard deviation of 10 kg. What is the probability that a randomly-selected ostrich will have a mass between 124 kg and 154 kg? a). 0.15%
b). 99.85%
c). 49.85%
d). 47.5%
Copyright © Cengage Learning. All rights reserved.
7|9
The Empirical Rule The masses of the adult ostriches at the Vilas Zoo are normally distributed with a mean of 124 kg and a standard deviation of 10 kg. What is the probability that a randomly-selected ostrich will have a mass between 124 kg and 154 kg? a). 0.15%
b). 99.85%
c). 49.85%
d). 47.5%
Copyright © Cengage Learning. All rights reserved.
7 | 10
Raw Scores and z Scores
Copyright © Cengage Learning. All rights reserved.
7 | 11
Raw Scores and z Scores
For a distribution with = 10 and = 2.5, find the z score of the value x = 15.
a). z = 2
b). z = 12.5
c). z = 4
d). z = 3.16
Copyright © Cengage Learning. All rights reserved.
7 | 12
Raw Scores and z Scores
For a distribution with = 10 and = 2.5, find the z score of the value x = 15.
a). z = 2
b). z = 12.5
c). z = 4
d). z = 3.16
Copyright © Cengage Learning. All rights reserved.
7 | 13
Distribution of z-Scores • If the original x values are normally distributed, so are the z scores of these x values. – µ=0 – σ=1
Copyright © Cengage Learning. All rights reserved.
7 | 14
Using the Standard Normal Distribution There are extensive tables for the Standard Normal Distribution. • We can determine probabilities for normal distributions: 1) Transform the measurement to a z score. 2) Utilize Table 3 of the Appendix.
Copyright © Cengage Learning. All rights reserved.
7 | 15
Using the Standard Normal Table • Table 3(a) gives the cumulative area for a given z value. • When calculating a z Score, round to 2 decimal places. • For a z Score less than –3.49, use 0.000 to approximate the area. • For a z Score greater than 3.49, use 1.000 to approximate the area.
Copyright © Cengage Learning. All rights reserved.
7 | 16
Area to the Left of a Given z Value
Copyright © Cengage Learning. All rights reserved.
7 | 17
Area to the Right of a Given z Value
Copyright © Cengage Learning. All rights reserved.
7 | 18
Area Between Two z Values
Copyright © Cengage Learning. All rights reserved.
7 | 19
Using a z Table Using Table 3 in the Appendix , find the probability that z > 0.9. a). 0.22
b). 0.09
c). 0.65
d). 0.18
Copyright © Cengage Learning. All rights reserved.
7 | 20
Using a z Table Using Table 3 in the Appendix , find the probability that z > 0.9. a). 0.22
b). 0.09
c). 0.65
d). 0.18
Copyright © Cengage Learning. All rights reserved.
7 | 21
Normal Probability Final Remarks • The probability that z equals a certain number is always 0. – P(z = a) = 0 • Therefore, < and ≤ can be used interchangeably. Similarly, > and ≥ can be used interchangeably. – P(z < b) = P(z ≤ b) – P(z > c) = P(z ≥ c)
Copyright © Cengage Learning. All rights reserved.
7 | 22
Inverse Normal Distribution • Sometimes we need to find an x or z that corresponds to a given area under the normal curve. – In Table 3, we look up an area and find the corresponding z.
Copyright © Cengage Learning. All rights reserved.
7 | 23
Finding z Corresponding to a Given Area A (0 < A < 1) Left-tail case: the given area is to the left of z.
Look up the number A in the body of the table and use the corresponding z value. Copyright © Cengage Learning. All rights reserved.
7 | 24
Finding z Corresponding to a Given Area A (0 < A < 1) Right-tail case: the given area is to the right of z.
Look up the number 1 – A in the body of the table and use the corresponding z value. Copyright © Cengage Learning. All rights reserved.
7 | 25
Finding z Corresponding to a Given Area A (0 < A < 1) Center-tail case: the given area is symmetric and centered above z = 0.
Look up the number (1 – A)/2 in the body of the table and use the corresponding ±z value. Copyright © Cengage Learning. All rights reserved.
7 | 26
Inverse Normal Distribution Using Table 3 in the Appendix, find the range of z scores, centered about the mean, that contain 70% of the probability.
a). –1.04 to 1.04
b). –2.17 to 2.17
c). –0.30 to 0.30
d). –0.52 to 0.52
Copyright © Cengage Learning. All rights reserved.
7 | 27
Inverse Normal Distribution Using Table 3 in the Appendix, find the range of z scores, centered about the mean, that contain 70% of the probability.
a). –1.04 to 1.04
b). –2.17 to 2.17
c). –0.30 to 0.30
d). –0.52 to 0.52
Copyright © Cengage Learning. All rights reserved.
7 | 28
Critical Thinking – How to tell if data follow a normal distribution? • Histogram – a normal distribution’s histogram should be roughly bell-shaped. • Outliers – a normal distribution should have no more than one outlier
Copyright © Cengage Learning. All rights reserved.
7 | 29
Critical Thinking – How to tell if data follow a normal distribution? • Skewness –normal distributions are symmetric. Use the Pearson’s index:
3( x median ) Pearson’s index = s A Pearson’s index greater than 1 or less than –1 indicates skewness. • Normal quantile plot – using a statistical software (see the Using Technology feature.) Copyright © Cengage Learning. All rights reserved.
7 | 30
Terms, Statistics & Parameters • Terms: Population, Sample, Parameter, Statistics
Copyright © Cengage Learning. All rights reserved.
7 | 31
Why Sample? • If time and resources are limited, we take samples to learn about the population.
Copyright © Cengage Learning. All rights reserved.
7 | 32
Types of Inference 1) Estimation: We estimate the value of a population parameter. 2) Testing: We formulate a decision about a population parameter. 3) Regression: We make predictions about the value of a statistical variable.
Copyright © Cengage Learning. All rights reserved.
7 | 33
Sampling Distributions • To evaluate the reliability of our inference, we need to know about the probability distribution of the statistic we are using. • Typically, we are interested in the sampling distributions for sample means and sample proportions.
Copyright © Cengage Learning. All rights reserved.
7 | 34
The Central Limit Theorem (Normal) • If x is a random variable with a normal distribution, mean = µ, and standard deviation = σ, then the following holds for any sample size:
(n is the sample size)
Copyright © Cengage Learning. All rights reserved.
7 | 35
The Standard Error • The standard error is just another name for the standard deviation of the sampling distribution.
Copyright © Cengage Learning. All rights reserved.
7 | 36
The Central Limit Theorem (Any Distribution) • If a random variable has any distribution with mean = µ and standard deviation = σ, the sampling distribution of x will approach a normal distribution with mean = µ and standard deviation = n as n increases without limit.
Copyright © Cengage Learning. All rights reserved.
7 | 37
Sample Size Considerations • For the Central Limit Theorem (CLT) to be applicable: – If the x distribution is symmetric or reasonably symmetric, n ≥ 30 should suffice. – If the x distribution is highly skewed or unusual, even larger sample sizes will be required. – If possible, make a graph to visualize how the sampling distribution is behaving.
Copyright © Cengage Learning. All rights reserved.
7 | 38
Critical Thinking • Bias – A sample statistic is unbiased if the mean of its sampling distribution equals the value of the parameter being estimated. • Variability – The spread of the sampling distribution indicates the variability of the statistic.
Copyright © Cengage Learning. All rights reserved.
7 | 39
Normal Approximation to the Binomial
Copyright © Cengage Learning. All rights reserved.
7 | 40
Normal Approximation to the Binomial A fair coin is flipped 200 times and the number of heads, x, is counted. Find the normal approximation of the standard deviation for this experiment. a). 50
b). 7.07
Copyright © Cengage Learning. All rights reserved.
c). 10
d). 100
7 | 41
Normal Approximation to the Binomial A fair coin is flipped 200 times and the number of heads, x, is counted. Find the normal approximation of the standard deviation for this experiment. a). 50
b). 7.07
Copyright © Cengage Learning. All rights reserved.
c). 10
d). 100
7 | 42
Continuity Correction
Copyright © Cengage Learning. All rights reserved.
7 | 43
View more...
Comments