Section 6.4 -6.5 Notes complete stats
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Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
Section 6-4 The Central Limit Theorem
In this section we introduce and apply the central limit theorem.
The central limit theorem allows us to use a normal distribution for some very meaningful and important applications.
Central Limit Theorem
1. The sampling distribution of x
(or distribution of sample means) will, as the sample size increases, approach a normal distribution. (
n > 30 )
2. The sampling distribution of x
can be approximated by the population mean.
3. The standard deviation of all sample means can be approximated by σ
x
=
σ
√
n
Given
1. The original population has with mean (
μ
)
and standard deviation (
σ
)
.
2. Simple random samples of the same size n
are selected from the population.
Requirements: Population has a normal distribution or n
> 30:
This means we can still use our table if we adjust our calculation for z!
Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
NOTES
Original population is not normally distributed and n
≤ 30:
The distribution of x
cannot be approximated well by a normal distribution, and the methods of this section do not apply.
Applying the Central Limit Theorem
1. Working with an individual
value from a normally distributed population, use z
=
x
−
μ
σ
.
2. Working with the mean for some sample
(or group), be sure to use the value of σ
x
=
σ
√
n
for the standard deviation of the sample mean. So, use z
=
x
−
μ
(
σ
√
n
)
Example 1
Use Example 2
Children between the ages of 2 and 5 watch an average of 25 hours of TV per week with a standard deviation of
3 hours. If a sample of 20 children are selected, find the probability that the mean number of hours watched per week will be greater than 26.3? Variables in this problem: µ = mean = ______ n = sample size = ______ σ = standard deviation = _______ 𝑥
̅ = mean of sample = _______
Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
Example 3
A certain kind of bacteria exists in all water. Let x be the bacteria count per milliliter of water. The health department has found that if the water is not contaminated, then x has a distribution that is more or less mound-
shaped and symmetrical. The mean of x is µ = 3000 and the standard deviation is σ = 200. The city health inspector takes 64 random samples of water from the city public water system each day. Assuming the water is not contaminated, what is the probability that 𝑥
̅ is less than 2925?
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Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
Example 4
The average sales price of a single-family house in the United States is $176,800. You randomly select 12 single-family houses. What is the probability that the mean sales price is more than $160,000? Assume that the sales prices are normally distributed with a standard deviation of $50,000.
Example 5
: Is it unusual? The weights of ice cream cartons produced by a manufacturer are normally distributed with a mean weight of 10 ounces and a standard deviation of 0.5 ounce.
Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
(a)
What is the probability that a randomly selected carton has a weight greater than 10.21 ounces? Does this seem unusual? (b) You select 25 cartons. What is the probability that their mean weight is greater than 10.21 ounces? Does this seem unusual?
Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
Example 6:
Is it unusual? A machine is set to fill paint cans with a mean of 128 ounces and a standard deviation of 0.2 ounce. A random sample of 40 cans has a mean of 127.9 ounces. Does the machine need to be reset?
Example 7 Southwest Airline Seats
Southwest Airlines currently has a seat width of 17 in. Men have hip breadths that are normally distributed with a mean of 14.4 in. and a standard deviation of 1.0 in. (based on anthropometric survey data from Gordon, Churchill, et al.). (a)
Find the probability that if an individual man is randomly selected, his hip breadth will be greater 17 in.
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Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
(b) Southwest Airlines uses a Boeing 737 for some of its flights, and that aircraft seats 122 passengers. If the plane is full with 122 randomly selected men, find the probability that these men have a mean hip breadth greater than 17 in.
(c)
Which result should be considered for any changes in seat design: the result from part (a) or part (b)?
Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
Section 6.5 Assessing Normality
In this section we present criteria for determining whether the requirement of a normal distribution is satisfied. The criteria involve (1) visual inspection of a histogram to see if it is roughly bell-shaped; (2) identifying any outliers; and (3) constructing a normal quantile plot.
•
Normal Quantile Plot
A normal quantile plot (or normal probability plot
) is a graph of points (
x
, y
) where each x value is from the original set of sample data, and each y value is the corresponding z score that is expected from the standard normal distribution.
Procedure for Determining Whether It Is Reasonable to Assume That Sample Data Are from a Population Having a Normal Distribution
1.
Histogram:
Construct a histogram. If the histogram departs dramatically from a bell shape, conclude that the data do not have a normal distribution.
2.
Outliers:
Identify outliers. If there is more than one outlier present, conclude that the data might not have a normal distribution.
3.
Normal quantile plot:
If the histogram is basically symmetric and the number of outliers is 0 or 1, use
technology to generate a
normal quantile plot.
Apply the following criteria to determine whether the distribution is normal. (These criteria can be used loosely for small samples, but they should be used more strictly for large samples.)
Normal Distribution: The population distribution is normal if the pattern of the points is reasonably close to a straight line and the points do not show some systematic pattern that is not a straight-line pattern.
Not a Normal Distribution: The population distribution is not
normal if either or both of these two conditions applies:
•
The points do not lie reasonably close to a straight line.
•
The points show some systematic pattern that is not a straight-line pattern.
Normal Example
Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
Normal: The first case shows a histogram of IQ scores that is close to being bell-shaped suggesting a normal distribution. The corresponding normal quantile plot shows points that are reasonably close to a straight-line pattern, and the points do not show any other systematic pattern that is not a straight line. It is safe to assume that these IQ scores are from a population that has a normal distribution
Uniform Example
Uniform: The second case shows a histogram of data having a uniform (flat) distribution. The corresponding normal quantile plot suggests that the points are not normally distributed. Although the pattern of points is reasonably close to a straight-line pattern, there is another
systematic pattern that is not a straight-line pattern. We conclude that these sample values are from a population having a distribution that is not normal.
Skewed Example
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Math 1442 (Statistics for Life)
Ch 6
Sec 6.4-6.5
Skewed: The third case shows a histogram of the amounts of rainfall (in inches) in Boston for every Monday during one year. The shape of the histogram is skewed to the right, not bell-shaped. The corresponding normal quantile plot shows points that are not at all close to a straight-line pattern. These rainfall amounts are from a population having a distribution that is not normal.
Tools for Determining Normality
•
Histogram / Outliers: If the requirement of a normal distribution is not too strict, simply look at a histogram and find the number of outliers. If the histogram is roughly bell-shaped and the number of outliers is 0 or 1, treat the population as if it has a normal distribution.
•
Normal Quantile Plot: Normal quantile plots can be difficult to construct on your own, but they can be generated with a TI-83/84 Plus calculator or suitable software, such as Statdisk, Minitab, Excel, or StatCrunch.