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Lesson 10 - R
Summary of Hypothesis Testing
Objectives
• Review Hypothesis Testing
Hypothesis Testing
• The process of hypothesis testing is very
similar across the testing of different
parameters
• The major steps in hypothesis testing are
– Formulate the appropriate null and alternative
hypotheses
– Calculate the test statistic
– Determine the appropriate critical value(s)
– Reach the reject / do not reject conclusions
Similarities in hypothesis test
processes
Parameter
Mean
(σ known)
Mean
(σ unknown)
Proportion
Variance
Std Dev
H0 :
μ = μ0
μ = μ0
p = p0
σ2 = σ02
σ = σ0
(2-tailed) H1:
μ ≠ μ0
μ ≠ μ0
p ≠ p0
σ2 ≠ σ02
σ ≠ σ0
(L-tailed) H1:
μ < μ0
μ < μ0
p < p0
σ < σ02
σ < σ0
(R-tailed) H1:
μ > μ0
μ > μ0
p > p0
σ > σ02
σ > σ0
Test statistic
Difference
Difference
Difference
Ratio
Ratio
Critical value
Normal
Student t
Normal
Chi-square
Chi-square
Chapter 10 – Section 1
If a researcher wishes to test a claim that the
average weight of a white rhinoceros is 5,000
lbs, then she should state a null hypothesis of
1) H1: Average weight = 5,000 pounds
2) H0: Average weight = 5,000 pounds
3) H0: Average weight ≠ 5,000 pounds
4) H0 + H1: Average weight = 5,000 pounds
Chapter 10 – Section 1
If the hypotheses for a test are
H0: μ = 20 seconds
H1: μ < 20 seconds
then an example of a Type I error occurs when
1) μ = 20 seconds and we did not reject H0
2) μ = 15 seconds and we rejected H0
3) μ = 25 seconds and we did not reject H0
4) μ = 20 seconds and we rejected H0
Chapter 10 – Section 2
The classical approach rejects the null
hypothesis H0: μ = 20 when
1) The sample mean is far (too many standard
deviations) from 20
2) The sample mean is not equal to 20
3) The sample mean is close (too few standard
deviations) to 20
4) The sample mean is equal to 20
Chapter 10 – Section 2
In the P-value approach, relatively small values
of the P-value correspond to situations where
1) The classical approach does not apply
2) The null hypothesis H0 must be accepted
3) The null hypothesis H0 must be rejected
4) The probability of obtaining such a sample mean
is relatively small
Chapter 10 – Section 3
When the population standard deviation σ is
not known, then we should perform hypothesis
tests using
1) The alternative hypothesis
2) The t-distribution
3) The normal distribution
4) The Type II Error
Chapter 10 – Section 3
In testing a claim regarding a population mean
with σ is unknown, we
1) May use only the classical approach with the
t-distribution
2) May use only the P-value approach with the
t-distribution
3) May use either the classical approach or the Pvalue approach with the t-distribution
4) May use either standard normal distribution with
the t-distribution
Chapter 10 – Section 4
A possible null hypothesis for testing a claim
regarding a population proportion is
1) H0: Mean Weight of Dogs = 20 kgs
2) H0: Standard Deviation of Weight of Dogs = 8 kgs
3) H0: Proportion of Dogs Weighs 30 kgs
4) H0: Proportion of Dogs that weigh < 30 kgs = 0.30
Chapter 10 – Section 4
Tests of a claim about a population proportion use
1) The normal model, or the binomial probability
distribution if the sampling distribution is not normal
2) Always the normal model
3) Always the Type II model
4) The t-distribution, or the sampling distribution if the
sample size is too small
Chapter 10 – Section 5
The test of a claim about a population standard
deviation uses the
1) Normal distribution
2) The t-distribution
3) The chi-square distribution
4) All of the above
Chapter 10 – Section 5
If a sample size n is 65, then a test of a claim
about a population standard deviation uses
1) A normal distribution with mean 65
2) A normal distribution with standard deviation 64
3) A chi-square distribution with 65 degrees of
freedom
4) A chi-square distribution with 64 degrees of
freedom
Chapter 10 – Section 6
To determine the appropriate hypothesis test
to perform, we should
1) Consider which P-value we wish to obtain
2) Consider which type of parameter we are
analyzing
3) Consider whether the null hypothesis is known
or unknown
4) All of the above
Chapter 10 – Section 7
If the hypotheses for a test are
H0: μ = 20 seconds
H1: μ < 20 seconds
then an example of a Type II error occurs when
1) μ = 25 seconds and we did not reject H0
2) μ = 15 seconds and we rejected H0
3) μ = 15 seconds and we did not reject H0
4) μ = 20 seconds and we rejected H0
Chapter 10 – Section 7
A large power for a test occurs when
1) The Type II error β is small
2) The probability of failing to reject the null
hypothesis, when the alternative hypothesis is
true, is small
3) Distinguishing between the null hypothesis and
the alternative hypothesis is relatively clear with
the data
4) All of the above
Hypothesis Testing
H0: The status quo, what was done before, what we are
trying to disprove
H1: The new item, the new study results
Test Statistics:
Test μ σ unknown
x-μ
Z0 = ---------σ / √n
Test μ σ known
x-μ
t0 = ---------s / √n
p^ - p
Z0 = ---------p(1-p)
-------n
Test σ
n s²
χ²0 = -------σ²
Test population prop
Critical Values: (left, two, right tailed tests)
Zc = Zα, 1-α/2, 1-α; tc = tα, 1-α/2, 1-α/n-1; χ²c = χ²1-α, 1-α/2, α/n-1
Conclusion:
If Zc < Z0, tc < t0, p < α, or χ²c < χ²0 then Reject H0
Otherwise we Fail to Reject (FTR)
Hypothesis Testing Methods
Q0
FTR
• Classical
Qα
– More standard deviations away from mean
– Probability of getting a more extreme value
– Within the interval
Q0
FTR
• P-Value
• Confidence Interval
Rej H0
FTR
Rej H0
LB
Qα
Rej H0
Rej H0
UB
Requirements to Check
• Mean, σ Known
– Simple Random Sample (SRS)
– Normal distribution
• Mean, σ unknown
– SRS
– No outliers and “normality” (normality plot)
• Population Proportion
– SRS
– n(p)(1-p) ≥ 10
– n ≤ 0.05N
(allows normal estimation of binomial)
(keeps it from being hypergeometric)
• Variance or Standard Deviation
– SRS
– Normal distribution
Hypothesis Test – Mean, σ Known
USAA Auto Insurance data base show the average miles driven is
12,200. A local rep, Sam, believes the residents of southwestern
Virginia drive more. He obtains a sample of 35 drivers whose
average was 12,895.9. Using USAA’s database σ = 3800 miles.
Test his claim at the α = 0.01 level.
H0: μ0 = 12,200 (drivers in southwestern VA drive the same as elsewhere)
H1: μ0 > 12,200 (drivers in southwestern VA drive more than elsewhere)
x-bar = 12,895.9
μ0 = 12,200
σ = 3800
n = 35
α = 0.01
X-bar – μ
Z0 = --------------- = 1.083 and p = 0.13931 (from calculator)
σ / √n
Critical Values: Zc = 2.326
Confidence Interval (CI) [11241, 14550]
Conclusion: Since Z0 < Zc (μ0 in CI or p > α), we fail to reject H0 and conclude
that we don’t have sufficient evidence to say SWVA drivers drive more.
Hypothesis Test – Mean, σ Unknown
A high school principal believes that the new attendance policy
has reduced the average number of tardies among the habitual
tardy students. He samples 40 of his habitual tardy students and
determines that their average tardies was 16.8 with a standard
deviation of 4.7. He wants you to test at the α = 0.1 level to see if
the average number of tardies was less than the historic mean of
18.1.
H0: μ0 = 18.1 (habitual tardiness remained the same)
H1: μ0 < 18.1 (habitual tardiness decreased)
x-bar = 16.8
μ0 = 18.1
σ = 4.7
n = 40
α = 0.1
X-bar – μ
t0 = --------------- = -1.7493 and p = 0.04405 (from calculator)
s / √n
Critical Values: tc = -1.304
Confidence Interval (CI) [15.548, 18.052]
Conclusion: Since t0 < tc (μ0 out of CI or p < α), we reject H0 and conclude that
the habitual tardiness has decreased.
Hypothesis Test – Population Proportion
In the 1990’s 65% of students at Virginia Tech thought that lying was
unethical. In a poll conduct last May in a simple random sample of
1005 Virginia Tech students, 704 responded that lying was unethical.
Is there evidence to indicate that the percentage of students who
believe that lying is unethical has increased at the α = 0.05 level.
H0: p0 = 0.65 (% who thought lying was unethical behavior is the same)
H1: p0 > 0.65 (% who thought lying was unethical behavior has increased)
p0 = 0.65
x = 704
n = 1005
α = 0.05
p-hat – p0
Z0 = --------------- = 3.356 and p = 0.0004 (from calculator)
√p0(1-p0)/n
Critical Values: Zc = -1.304
Confidence Interval (CI) [0.672, 0.728]
Conclusion: Since Z0 > Zc (p0 out of CI or p < α), we reject H0 and conclude that
the percentage who believe lying is unethical has increased.
Hypothesis Test – Population Variance
A snack bag of plain M&M’s has a mean number of M&M’s of 21.
The quality control people at M&M-Mars have published data on the
internet the claims the standard deviation of the number of M&Ms to
be under 0.75. A Stats class samples 11 snack bags of plain M*Ms
and determines that the standard deviation was 0.6404. Their
teacher wants to know if their sample standard deviation is smaller
than the advertised at the α = 0.05 level
H0: σ0 = 0.75 (the standard deviation of M&Ms in snack bags is the same)
H1: σ0 < 0.75 (the standard deviation of M&Ms in snack bags has decreased)
σ0 = 0.75
s = 0.6404
n = 11
α = 0.05
n s²
χ²0 = --------------- = 7.291 (by hand) p-value = 0.302 (by χ²cdf)
σ²0
Critical Values: χ²c = 3.940
Confidence Interval: NA
Conclusion: Since χ0 > χc (or p > α), we fail to reject H0 and conclude that there
is insufficient evidence that σ in plain M&M snack bags has decreased.
Summary and Homework
• Summary
– We can test whether sample data supports a
hypothesis claim about a population mean,
proportion, or standard deviation
– We can use any one of three methods
• The classical method
• The P-Value method
• The Confidence Interval method
– The commonality between the three methods is that
they calculate a criterion for rejecting or not
rejecting the test statistic
• Homework
– pg 511-513; 1, 2, 3, 7, 8, 12, 13, 14, 15, 17, 20, 37