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Exam 2 Review & 9.3 - 9.4Review
Cathy Poliak, [email protected] in Fleming 11c
Department of MathematicsUniversity of Houston
Lecture 12
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 1 / 43
Outline
1 Review of Topics
2 Review Questions
3 Estimating Regression Parameters
4 Inferences for β1
5 F-test
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 2 / 43
Review Questions
We wish to test the hypotheses H0 : µ = 13.5 versus Ha : µ > 13.5 atα = 0.02 significance level. From a random sample of 40 the samplemean is 15.9. Assume that the population standard deviation is 2.7.
1. Which test would be appropriate to use?a) z-test for proportions b) z-tests for means
c) t-test for means d) t-tests for proportions
2. Calculate the test statistic.a) 2.4 b) -5.62 c) 5.62 d) 0.89 e) 2.0537
3. Determine the p-value, approximately.a) 0 b) 0.02 c) 1.434 d) 4.68
4. What is our decision of the test?a) Reject H0 b) Fail to reject H0 c) Accept H0
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 3 / 43
Review Questions
For each of the following scenarios, determine if it is a) paired t-test orb) two sample t-test
5. The weight of 14 patients before and after open-heart surgery.
6. The smoking rates of 14 men measured before and after a stroke.
7. The number of cigarettes smoked per day by 14 men who havehad strokes compared with the number smoked by 14 men whohave not had strokes.
8. The photosynthetic rates of 10 randomly chosen Douglas-fir treescompared with 10 randomly chosen western red cedar trees.
9. The photosynthetic rate measured on 10 randomly chosen Sitkaspruce trees compared with the rate measured on the western redcedar growing next to each of the Sitka spruce trees.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 4 / 43
Dates of Exam and Covered Chapters
Dates: July 12 & 13
Chapters: 5, 6.5, 7, 8, & 10
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 5 / 43
Chapter 5 Continuous Random Variables
Density functions
Know how to calculate expected values and quantiles forcontinuous distributions.
Named distributionsI UniformI ExponentialI GammaI Normal
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Chapter 6 Sampling Distributions
Expected values and variances of X + Y
Applying the Central Limit Theorem
The sampling distribution of the sample means, X .
The sampling distribution of the proportions, p.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 7 / 43
Confidence Intervals
Know how to calculate the confidence intervals for mean (µ),proportions (p) and standard deviation (σ).
For mean if σ, population standard deviation is given: x ± z∗(σ√n
).
For mean if σ is not given: x ± t∗df
(s√n
).
For proportions p = Xn : p ± z∗
√p(1−p)
n .
For standard deviation: lcl =√
(n−1)s2
qchisq(1−α/2,n−1) and
ucl =√
(n−1)s2
qchisq(α/2,n−1)
For paired - t: xD ± t∗df
(sD√
n
)For two-sample t for means: (x1 − x2)± t∗ν
√s2
1n2
+ s2s
n2
For two-sample for proportions: (p1 − p2)± z∗√
p1(1−p1)n1
+ p2(1−p2)n2
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 8 / 43
Degrees of Freedom for Two-Sample T
df = ν =
(s2
1n1
+s2
2n2
)2
1n1−1
(s2
1n1
)2+ 1
n2−1
(s2
2n2
)2
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 9 / 43
Confidence Intervals
Know how the confidence interval changes as the confidence levelC changes and as the sample size changes.
Know how to interpret confidence intervals.
Know how to determine a sample size given confidence level andmargin of error.
I For means n >(
zα/2×σE
)2, where E = margin of error.
I For proportions n > p∗(1− p∗)(zα/2
E )2, where p∗ is some previousknowledge of the proportion if not known we use 0.5.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 10 / 43
Hypothesis Tests
Know how to set up null and alternative hypotheses.Be able to determine a rejection region, given the level ofsignificance (α).Calculate a test statistic.
I For one - sampleH0 : µ = µ0 H0 : µ = µ0 H0 : p = p0if σ is given if σ is unknown
Test Statistic z = x−µ0σ/√
n t = x−µ0s√
n z = p−p0√p0(1−p0)
n
I For two-sampleH0 : µ1 = µ2 H0 : p1 = p2 H0 : µD = 0
Test Statistic t = x1−x2√s21
n1+
s22
n2
z = p1−p2√p1(1−p1)
n1+
p2(1−p2)
n2
t = xD−µdsD/√
n
Be able to calculate the p-value and know to reject (RH0) or fail toreject (FRH0).Know the difference between Type I and Type II errors.Be able to make a conclusion in context of the problem.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 11 / 43
Go through the practice test in the "Online Assignments"
Go through the review problems:https://www.math.uh.edu/~cathy/Math3339/Tests/Test%202/test2_review_sp19.pdf
Any questions about the material post on the discussion board.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 12 / 43
Example, Continuous Random Variable
Suppose a continuous random variable X has the following pdf:
f (x) =
{1ax 0 ≤ x ≤ 20 otherwise
1. Find the value of a that makes this function a valid pdf.
2. Determine the expected value of X.
3. Find P(X ≤ 1)
4. Determine the CDF.
5. Find the value of c such that P(X ≤ c) = 0.04.
6. Determine Q3, the third quartile.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 13 / 43
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 14 / 43
Example, Another Continuous Distribution
Suppose a continuous random variable X has the following pdf:
f (x) =
{13e−
13 x x ≥ 0
0 otherwise
1. Determine P(X ≤ 3)
2. Determine E(X)3. Determine SD(X)
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 15 / 43
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 16 / 43
Cumulative Functions
Let X be the amount of time (in hours) the wait is to get a table atrestaurant. Suppose the cdf is represented by
F (x) =
0 x < 0x2
9 0 ≤ x ≤ 31 x > 3
1. Determine P(1.5 ≤ x ≤ 4).2. Determine µ = E(X ).3. Determine σ, that is the standard deviation.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 17 / 43
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 18 / 43
Example of Confidence Interval
The average height of students at UH from an SRS of 17 studentsgave a standard deviation of 2.9 feet. Construct a 95% confidenceinterval for the standard deviation of the height of students at UH.Assume normality for the data.a) (1.160, 9.413)b) (6.160, 11.413)c) (4.160, 9.413)d) (1.660, 8.413)e) (2.160, 4.413)f) None of the above
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 19 / 43
Which Test?
Identify the most appropriate test to use for the following situations.1. A national computer retailer believes that the average sales are
greater for salespersons with a college degree. A random sampleof 14 salespersons with a degree had an average weekly sale of$3542 last year, while 17 salespersons without a college degreeaveraged $3301 in weekly sales. The standard deviations were$468 and $642 respectively. Is there evidence to support theretailer’s belief?
2. Quart cartons of milk should contain at least 32 ounces. A sampleof 22 cartons was taken and amount of milk in ounces wasrecorded. We would like to determine if there is sufficient evidenceexist to conclude the mean amount of milk in cartons is less than32 ounces?
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 20 / 43
Which Test?
Identify the most appropriate test to use for the following situations.1. In a experiment on relaxation techniques, subject’s brain signals
were measured before and after the relaxation exercises. We wishto determine if the relaxation exercise slowed the brain waves.
2. A private and a public university are located in the same city. Forthe private university, 1046 alumni were surveyed and 653 saidthat they attended at least one class reunion. For the publicuniversity, 791 out of 1327 sampled alumni claimed they haveattended at least one class reunion. Is the difference in thesample proportions statistically significant?
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 21 / 43
Least-Squares regression
The least-squares regression line (LSRL) of Y on X is the linethat makes the sum of the squares of the vertical distances of thedata points from the line as small as possible.The linear regression model is: Y = β0 + β1x + ε
I Y is dependent variable (response).I x is the independent variable (explanatory).I β0 is the population intercept of the line.I β1 is the population slope of the line.I ε is the error term which is assumed to have mean value 0. This is
a random variable that incorporates all variation in the dependentvariable due to factors other than x .
I The variability: σ of the response y about this line. More precisely,σ is the standard deviation of the deviations of the errors, εi in theregression model.
We will gather information from a sample so we will have the leastsquares estimates model: Y = β0 + β1x .
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 22 / 43
Principle of Least Squares
The vertical deviation of the point (xi , yi) from the line y = b0 + b1x is
hieght of point − height of line = yi − (b0 + b1xi)
The sum of the square vertical deviations from the points(x1, y1), (x2, y2), . . . , (xn, yn) to the line is then
f (b0,b1) =n∑
i=1
[yi − (b0 + b1xi)]2
The point estimates of β0 and β1, denoted by β0 and β1 and called theleast squares estimates, are those values that minimize f (b0,b1).
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 23 / 43
Estimating the Regression Parameters
In the simple linear regression setting, we use the slope b1 andintercept b0 of the least-squares regression line to estimate theslope β1 and intercept β0 of the population regression line.The standard deviation, σ, in the model is estimated by theregression standard error
s =
√∑(yi − yi)2
n − 2=
√∑all residuals2
n − 2
Recall that yi is the observed value from the data set and yi is thepredicted value from the equation.In R s is the called the Residual Standard Error in the lastparagraph of the summary.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 24 / 43
The Least - Squares Estimates
Recall ei = observed Y - predicted Y is the i th residual. Think of itas an estimate of the unobservable true random error εi .
The method of least squares selects estimators β0 and β1 thatminimizes the residual sum of squares:
SS(resid) = SSE =n∑
i=1
e2i =
n∑i=1
(Yi − Yi
)Where the estimate of the slope coefficient β1 is:
β1 =
∑(xi − x)(yi − y)∑
(xi − x)2 =Sxy
Sxx
The estimate for the intercept β0 is:
β0 = y − β1x
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 25 / 43
Example
The marketing manager of a supermarket chain would like to use shelfspace to predict the sales of coffee. A random sample of 12 stores isselected, with the following results.
Store Shelf Space (ft) Weekly Sales (# sold)1 5 1602 5 2203 5 1404 10 1905 10 2406 10 2607 15 2308 15 2709 15 280
10 20 26011 20 29012 20 310
The correlation coefficient is r = 0.827.Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 26 / 43
Least-Squares
5 10 15 20
150
200
250
300
Shelf Space(feet)
Num
ber
Sol
d
> plot(shelf$space,shelf$sold)> shelf.lm=lm(shelf$sold~shelf$space)> abline(shelf.lm)
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 27 / 43
Finding Equation for Coffee Sales
Given these values determine the least-square regression line (LSRL)equation for predicting number sold based on shelf space.
Explanatory variable: Shelf space (X ) x = 12.5 feet, sx = 5.83874feet.Response variable: Sales (Y ) y = 237.5 units sold, sy = 52.2451units sold.Correlation: r = 0.827
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 28 / 43
R code
> shelf.lm=lm(sold~space)> summary(shelf.lm)Call:lm(formula = sold ~ space)
Residuals:Min 1Q Median 3Q Max-42.00 -26.75 5.50 21.75 41.00
Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 145.000 21.783 6.657 5.66e-05 ***space 7.400 1.591 4.652 0.000906 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 30.81 on 10 degrees of freedomMultiple R-squared: 0.6839,Adjusted R-squared: 0.6523F-statistic: 21.64 on 1 and 10 DF, p-value: 0.0009057
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 29 / 43
Determining if the Model is Good
For the sample we can use R2 and the residuals to determine ifthe equation is a good way of predicting the response variable.
Another way to determine if this equation is a good way ofpredicting the response variable is to determine if the explanatoryvariable is needed (significant) in the equation.
These tests of significance and confidence intervals in regressionanalysis are based on assumptions about the error term ε.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 30 / 43
Assumptions about the error term ε
1. The error term ε is a random variable with a mean or expectedvalue of zero, that is E(ε) = 0, an estimate for ε is the residuals foreach value of the X-variable.
residual = observed y− predicted y
2. The variance of ε, denoted by σ2, is the same for all values of x .The estimate for σ2 is s2 = MSE = SSE
n−2 =∑
(yi−yi )2
n−2 .
3. The values of ε are independent.
4. The error term ε is a normally distributed random variable.
5. The residual plots help us assess the fit of a regression line anddetermine if the assumptions are met.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 31 / 43
Definitions of Regression Output
1. The error sum of squares, denoted by SSE is
SSE =∑
(yi − yi)2
2. A quantitative measure of the total amount of variation in observedvalues is given by the total sum of squares, denoted by SST .
SST =∑
(yi − y)2
3. The regression sum of squares, denoted SSR is the amount oftotal variation that is explained by the model
SSR =∑
(yi − y)2
4. The coefficient of determination, r2 is given by
r2 =SSRSST
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 32 / 43
Finding these values using R
> anova(shelf.lm)Analysis of Variance Table
Response: soldDf Sum Sq Mean Sq F value Pr(>F)space 1 20535 20535 21.639 0.0009057 ***Residuals 10 9490 949---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 33 / 43
Conditions for regression inference
The sample is an SRS from the population.
There is a linear relationship in the population.
The standard deviation of the responses about the population lineis the same for all values of the explanatory variable.
The response varies Normally about the population regressionline.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 34 / 43
t Test for Significance of β1
HypothesisH0 : β1 = β10 versus Ha : β1 6= β10
Usually β10 = 0Test statistic
t =observed− hypothesized
standard deviation of observedobserved = b1
hypothesized = 0
standard error = SEb1 =s√∑
(xi − x)2
With degrees of freedom df = n − 2.P-value: based on a t distribution with n − 2 degrees of freedom.Decision: Reject H0 if p-value ≤ α.Conclusion: If H0 is rejected we conclude that the explanatoryvariable x can be used to predict the response variable y .
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 35 / 43
Testing β1
1. We want to test: H0 : β1 = 0 versus Ha : β1 6= 0 for the coffeesales.
2. Test statistic: t = (7.4−0)1.591 = 4.652
3. P-value: 2 ∗ P(T > 4.652) = 2 ∗ 0.00425 = 0.00906
4. Decision: Reject the Null hypothesis
5. Conclusion: β1 is significantly not zero, thus shelf space can beused to predict the number of units sold.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 36 / 43
R code
> shelf.lm=lm(sold~space)> summary(shelf.lm)Call:lm(formula = sold ~ space)
Residuals:Min 1Q Median 3Q Max-42.00 -26.75 5.50 21.75 41.00
Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 145.000 21.783 6.657 5.66e-05 ***space 7.400 1.591 4.652 0.000906 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 30.81 on 10 degrees of freedomMultiple R-squared: 0.6839,Adjusted R-squared: 0.6523F-statistic: 21.64 on 1 and 10 DF, p-value: 0.0009057
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 37 / 43
Height
Because elderly people may have difficulty standing to have theirheights measured, a study looked at predicting overall height fromheight to the knee. Here are data (in centimeters, cm) for five elderlymen:
Knee Height (cm) 57.7 47.4 43.5 44.8 55.2Overall Height(cm) 192.1 153.3 146.4 162.7 169.1
1. We want to test if the relationship of the measurement of theheight is more than double that of the measurement from the floorto the knee. Test H0 : β1 = 2 versus Ha : β1 > 2.
2. Give a conclusion for the relationship between using knee lengthto predict overall height.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 38 / 43
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 39 / 43
Confidence Intervals for β1
If we want to know a range of possible values for the slope we can usea confidence interval.
Remember confidence intervals are
estimate± t∗ × standard error of the estimate
Confidence interval for β1 is
b1 ± tα/2,n−2 × SEb1
Where t∗ is from table D with degrees of freedom n − 2 where n =number of observations.In R we can get this by confint(name.lm,level = 0.95).> confint(shelf.lm)2.5 % 97.5 %(Intercept) 96.464405 193.53560space 3.855461 10.94454
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 40 / 43
Determine a 90% Confidence Interval for CoffeeExample
From the output in R:
Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 145.000 21.783 6.657 5.66e-05 ***space 7.400 1.591 4.652 0.000906 ***
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 41 / 43
F-distribution
The F distribution with ν1 degrees of freedom in the numeratorand ν2 degrees of freedom in the denominator is the distribution ofa random variable
F =U/ν1
V/ν2,
where U ∼ χ2(df = ν1) and V ∼ χ2(df = ν2) are independent.That F has this distribution is indicated by F ∼ F (ν1, ν2).Notice U = SSR
σ2 ∼ χ2(df = 1) and V = SSEσ2 ∼ χ2(df = n − 2) are
independent.Let MSE = SSE/(n − 2) and MSR = SSR/1. Then
F =MSRMSE
=SSR/1
SSE/(n − 2)∼ F (1,n − 2)
Then we can use the F-distribution to test the hypothesisH0 : β1 = 0 versus Ha : β1 6= 0.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 42 / 43
F-test for Shelf Space
Analysis of Variance Table
Response: soldDf Sum Sq Mean Sq F value Pr(>F)space 1 20535 20535 21.639 0.0009057 ***Residuals 10 9490 949---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Note: F = t2 and the p-value is the same.
Cathy Poliak, Ph.D. [email protected] Office in Fleming 11c (Department of Mathematics University of Houston )Review Lecture 12 43 / 43