Statistics Review – Part I Topics – Z-values – Confidence Intervals – Hypothesis Testing – Paired Tests – T-tests – F-tests 1

Statistics Review – Part I

Topics– Z-values– Confidence Intervals– Hypothesis Testing– Paired Tests– T-tests– F-tests

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Statistics References

References used in class slides:

1. Sullivan III, Michael. Statistics: Informed Decisions Using Data, Pearson Education, 2004.

2. Gitlow, et. al Six Sigma for Green Belts and Champions, Prentice Hall, 2004.

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Relative frequency histograms that are symmetric and bell-shaped are said to have the shape of a normal curve.

Sampling and the Normal Distribution

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If a continuous random variable is normally distributed or has a normal probability distribution, then a relative frequency histogram of the random variable has the shape of a normal curve (bell-shaped and symmetric).


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• Suppose that the mean normal sugar level in the population is 0=9.7mmol/L with std. dev. =2.0mmol/L - you want to see whether diabetics have increased blood sugar level

• Sample n=64 individuals with diabetes mean is 0=13.7mmol/L with std. dev. =2.0mmol/L

• How do you compare these values?– Standardize!

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Reading z-scores


• Standardization:– Using Z-tables to evaluate sample means

– Puts samples on the same scale• Subtract mean and divide by standard deviation

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• Why do we standardize?– Enables the comparison of populations/ samples using a

standardized set of values

– Recall

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The table gives the area under the standard normal curve for values to the left of a specified Z-score, zo, as shown in the figure.


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– Population Mean=10, Standard Deviation=5

– What is the likelihood of a sample (n=16) having a mean greater than 12 (standard deviation = 5)?

– What is the likelihood of a sample (n=16) having a mean of less than 8 (standard deviation = 5)?

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Notation for the Probability of a Standard Normal Random Variable:

P(a < Z < b) represents the probability a standard normal random variable is between a and b

P(Z > a) represents the probability a standard normal random variable is greater than a.

P(Z < a) represents the probability a standard normal random variable is less than a.


• Before using Z-tables, need to assess whether the data is normally distributed

• Different ways– Histogram– Probability plot

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Normal Probability Plots:

Histogram CumulativeDistribution

NormalProbability Plot


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Normal Probability Plots:

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.25

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140145150155160Thickness (Ang)

Fat pencil test to detect normality


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Shapes of Normal Probability Plots:

.01

.05

.10

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.50

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.90

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.99

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.045.050.055.060.065.070.075

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0 1 2 3 4 5

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100.0101.0102.0103.0104.0

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Normal Probability Plots vs Box plots:

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1 2

Eqp

.01 .05 .10 .25 .50 .75 .90 .95 .99

-3 -2 -1 0 1 2 3

Normal Quantile

ThIckness

12000

13000

CLCPCRLPNPRPTP

site

.01.05.10.25.50.75.90.95.99

-3-2-10123

Normal Quantile

• If distribution of data is “approximately” normally distributed, use Z-tables to determine likelihood of events

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• Can also “flip” Z-scores to determine the ‘highest’ or ‘lowest’ acceptable sample mean

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– Point estimate: value of a statistic that estimates the value of the parameter.

– Confidence interval estimate: interval of numbers along

with a probability that the interval contains the unknown parameter.

– Level of confidence: a probability that represents the percentage of intervals that will contain if a large number of repeated samples are obtained.

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Confidence Intervals

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The construction of a confidence interval for the population mean depends upon three factors: The point estimate of the population The level of confidence The standard deviation of the sample mean:

A 95% level if 100 confidence intervals were constructed, each based on a different sample from the same population, we would expect 95 of the intervals to contain the population mean.


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If a simple random sample from a population is normally distributed or the sample size is large, the distribution of the sample mean will be normal with:


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95% of all sample means are in the interval:

With a little algebraic manipulation, we can rewrite this inequality and obtain:


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Steps to constructing a confidence interval:

1. Verify normality if n<=30.

2. Determine /2, x-bar, .

3. Find z-score for /2.

4. Calculate upper and lower bound.


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Histogram for z


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Histogram for t


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Properties of the t Distribution1. The t distribution is different for different values of n.2. The t distribution is centered at 0 and is symmetric about 0.3. The area under the curve is 1. The area under the curve to the right of 0

= the area under the curve to the left of 0 = 1 / 2.4. As t increases and decreases without bound, the graph approaches, but

never equals, zero. 5. The area in the tails of the t distribution is a little greater than the area in

the tails of the standard normal distribution. This is due to using s as an estimate introducing more variability to the t statistic.

6. As the sample size n increases, the density of the curve of t approaches the standard normal density curve. The occurs due to the values of s approaching the values of sigma by the law of large numbers.


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EXAMPLE: Finding t-values

Find the t-value such that the area under the t distribution to the right of the t-value is 0.2 assuming 10 degrees of freedom.

Hint: find t0.20 with 10 degrees of freedom.


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EXAMPLE: Finding Chi-Square Values

Find the chi-square values that separate the middle 95% of the distribution from the 2.5% in each tail. Assume 18 degrees of freedom.


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EXAMPLE: Constructing a Confidence Interval about a Population Standard Deviation


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Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of one or more populations.

Selecting Hypothesis Testing methods – see next slides.

Hypothesis Testing

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The null hypothesis, denoted Ho (read “H-naught”), is a statement to be tested. The null hypothesis is assumed true until evidence indicates otherwise. In this chapter, it will be a statement regarding the value of a population parameter.

The alternative hypothesis, denoted, H1 (read “H-one”), is a claim to be tested. We are trying to find evidence for the alternative hypothesis. In this chapter, it will be a claim regarding the value of a population parameter.

Hypothesis Testing

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There are three ways to set up the null and alternative hypothesis:

1. Equal versus not equal hypothesis (two-tailed test)Ho: parameter = some valueH1: parameter some value

2. Equal versus less than (left-tailed test)Ho: parameter = some valueH1: parameter < some value

3. Equal versus greater than (right-tailed test)Ho: parameter = some valueH1: parameter > some value

Hypothesis Testing

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THREE WAYS TO STRUCTURE THE HYPOTHESIS TEST:

Hypothesis Testing

• Two-tailed test

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Hypothesis Testing

• One-Tailed Test

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Hypothesis Testing

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Four Outcomes from Hypothesis Testing

1. We could reject Ho when in fact H1 is true. This would be a correct decision.

2. We could not reject Ho when in fact Ho is true. This would be a correct decision.

3. We could reject Ho when in fact Ho is true. This would be an incorrect decision. This type of error is called a Type I error.

4. We could not reject Ho when in fact H1 is true. This would be an incorrect decision. This type of error is called a Type II error.

Hypothesis Testing

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For example, we might reject the null hypothesis if the sample mean is more than 2 standard deviations above the population mean. Why?

z 0 1 2

Area = 0.0228

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If the null hypothesis is true, then 1 - 0.0228 = 0.9772 = 97.72% of all sample means will be less than

Hypothesis Testing

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Because sample means greater than 2.88 are unusual if the population mean is 2.62, we are inclined to believe the population mean is greater than 2.62.

Hypothesis Testing

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Hypothesis Testing

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Step 1: A claim is made regarding the population mean. The claim is used to determine the null and alternative hypotheses. Again, the hypothesis can be structured in one of three ways:

Hypothesis Testing

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Hypothesis Testing

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The critical region or rejection region is the set of all values such that the null hypothesis is rejected.

Hypothesis Testing

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Hypothesis Testing

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Step 4: Compare the critical value with the test statistic:

Step 5: State the conclusion.

Hypothesis Testing

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A P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption the null hypothesis is true.

Hypothesis Testing

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Hypothesis Test Regarding μ with σ Known (P-values)

Hypothesis Testing

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Hypothesis Testing

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Step 3: Compute the P-value.

Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

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Properties of the t Distribution1. The t distribution is different for different values of n, the sample size.2. The t distribution is centered at 0 and is symmetric about 0.3. The area under the curve is 1. Because of the symmetry, the area under

the curve to the right of 0 equals the area under the curve to the left of 0 equals ½.

4. As t increases without bound, the graph approaches, but never equals, zero. As t decreases without bound the graph approaches, but never equals, zero.

5. The area in the tails of the t distribution is a little greater than the area in the tails of the standard normal distribution. This result is because we are using s as an estimate of which introduces more variability to the t statistic.

Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

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Step 3: Compute the test statistic

which follows Student’s t-distribution with n – 1 degrees of freedom.

Hypothesis Testing

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Step 5 : State the conclusion.

Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

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Hypothesis Test Regarding a Population Variance or Standard Deviation

If a claim is made regarding the population variance or standard deviation, we can use the following steps to test the claim provided

(1) the sample is obtained using simple random sampling(2) the population is normally distributed

Hypothesis Testing

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Step 1: A claim is made regarding the population variance or standard deviation. The claim is used to determine the null and alternative hypothesis. We present the three cases for a claim regarding a population standard deviation.

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Hypothesis Testing

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Hypothesis Testing

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Step 4: Compare the critical value with the test statistic.

Step 5: State the conclusion.

Hypothesis Testing

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A sampling method is independent when the individuals selected for one sample does not dictate which individuals are to be in a second sample. A sampling method is dependent when the individuals selected to be in one sample are used to determine the individuals to be in the second sample.

Dependent samples are often referred to as matched pairs samples.

Paired Testing

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EXAMPLE Independent versus Dependent Sampling

For each of the following, determine whether the sampling method is independent or dependent.

(a) A researcher wants to know whether the price of a one night stay at a Holiday Inn Express Hotel is less than the price of a one night stay at a Red Roof Inn Hotel. She randomly selects 8 towns where the location of the hotels is close to each other and determines the price of a one night stay.

(b) A researcher wants to know whether the newly issued “state” quarters have a mean weight that is different from “traditional” quarters. He randomly selects 18 “state” quarters and 16 “traditional” quarters. Their weights are compared.

Paired Testing

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In order to test the hypotheses regarding the mean difference, we need certain requirements to be satisfied.

• A simple random sample is obtained

• The sample data is matched pairs

• The differences are normally distributed or the sample size, n, is large (n > 30).

Paired Testing

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Paired Testing

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Paired Testing

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Paired Testing

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Paired Testing

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T-Tests

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T-Tests

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T-Tests

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T-Tests

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T-Tests

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T-Tests

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The degrees of freedom used to determine the critical value(s) presented in the last example are conservative.

Results that are more accurate can be obtained by using the following degrees of freedom:

T-Tests

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Lower Bound =

Upper Bound =

2 21 2

1 2 / 21 2

( )s s

x x tn n

2 21 2

1 2 / 21 2

( )s s

x x tn n

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Requirements for Testing Claims Regarding Two Population Standard Deviations

1. The samples are independent simple random samples.2. The populations from which the samples are drawn are normally distributed.

F-Tests

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Fisher's F-distribution

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Characteristics of the F-distribution

1. It is not symmetric. The F-distribution is skewed right.

2. The shape of the F-distribution depends upon the degrees of freedom in the numerator and denominator. This is similar to the distribution and Student’s t-distribution, whose shape depends upon their degrees of freedom.

3. The total area under the curve is 1.

4. The values of F are always greater than or equal to zero.

F-Tests

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F-Tests

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Is the critical F with n1 – 1 degrees of freedom in the numerator and n2 – 1 degrees of freedom in the denominator and an area of to the right of the critical F.

To find the critical F with an area of α to the left, use the following:

F-Tests

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Hypothesis Test Regarding the Two Means Population Standard Deviations

F-Tests

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F-Tests

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F-Tests

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F-Tests

Documents

Statistics Review – Part I Topics – Z-values – Confidence Intervals – Hypothesis Testing – Paired Tests – T-tests – F-tests 1