Download Inference for a Population Mean Statistics 111

The government claims that students earn an average of $4500 during their
summer break from studies. A random sample of students gave a sample
average of $3975 and a 95% CI was found to be ($3525,$4425). This
interval is interpreted to mean that:
1. If the study were to repeated many times, there is a 95% probability that
the true population mean summer earnings is not $4500 as the
government claims.
2. Because our specific CI does not contain the value $4500 there is a 95%
probability that the true population mean summer earnings is not $4500.
3. If we were to repeat our survey many times, then about 95% of all the CI
will contain the value $4500.
4. If we repeat our survey many times, then about 95% of our confidence
intervals will contain the true population value of the mean earnings of
students.
Statistics 111 - Lecture 13
Inference for a
Population Mean
Two-sample Tests for
difference in means
1
Comparing Two Samples
• Up to now, we have looked at inference for one sample
• Our next focus in this course is comparing the data from
two different samples
• For now, we will assume that these two different samples
are independent of each other and come from two
distinct populations
Population 1:1 , 1
Sample 1:
Population 2:2 , 2
, s1
Sample 2:
, s2
Blackout Baby Boom Revisited
• Nine months (Monday, August 8th) after Nov 1965
blackout, NY Times claimed an increased birth rate
• Already looked at single two-week sample: found no
significant difference from usual rate (430 births/day)
• What if we instead look at difference between weekends
and weekdays?
Sun Mon
Tue
Wed Thu
Fri
Sat
452
470
431
448
467
377
344
449
440
457
471
463
405
377
453
499
461
442
444
415
356
470
519
443
449
418
394
399
451
468
432
Weekdays
Weekends
2
Two-Sample Z test
• We want to test the null hypothesis that the two
populations have different means
• H0: 1 = 2 or equivalently, 1 - 2 = 0
• Two-sided alternative hypothesis: 1 - 2  0
• If we assume our population SDs 1 and 2 are known, we
can calculate a two-sample Z statistic:
• We can then calculate a p-value from this Z statistic using
the standard normal distribution
Two-Sample Z test for Blackout Data
• To use Z test, we need to assume that our pop. SDs are
known: 1 = 21.7 and 2 = 24.5
• We can then calculate a two-sided p-value for Z=7.5 using
the standard normal distribution
• From normal table, P(Z > 7.5) is less than 0.0002, so
our p-value = 2  P(Z > 7.5) is less than 0.0004
• We reject the null hypothesis at -level of 0.05 and
conclude there is a significant difference between birth
rates on weekends and weekdays
3
Two-Sample t test with unknown variances
• We still want to test the null hypothesis that the two
populations have equal means (H0: 1 - 2 = 0)
• If 1 and 2 are unknown, then we need to use the sample
SDs s1 and s2 instead, which gives us the two-sample T
statistic:
• The p-value is calculated using the t distribution, but what
degrees of freedom do we use?
• df can be complicated and often is calculated by software
• Simpler and more conservative: set degrees of freedom equal to
the smaller of (n1-1) or (n2-1)
Two-Sample t test for Blackout Data
• To use t test, we need to use our sample standard
deviations s1 = 21.7 and s2 = 24.5
• We need to look up the tail probabilities using the t
distribution
• Degrees of freedom is the smaller of n1-1 = 22
or n2-1 = 7
4
Two-Sample t test for Blackout Data
• From t-table with df = 7, we see that
P(T > 7.5) < 0.0005
• If our alternative hypothesis is two-sided, then we know
that our p-value < 2  0.0005 = 0.001
• We reject the null hypothesis at -level of 0.05 and
conclude there is a significant difference between birth
rates on weekends and weekdays
• Same result as Z-test, but we are a little more
conservative
5
Two-Sample Confidence Intervals
• In addition to two sample t-tests, we can also use the t
distribution to construct confidence intervals for the
mean difference
• When 1 and 2 are unknown, we can form the following
100·C% confidence interval for the mean difference 1 - 2
• The critical value tk* is calculated from a t distribution with
degrees of freedom k
• k is equal to the smaller of (n1-1) and (n2-1)
Confidence Interval for Blackout Data
• We can calculate a 95% confidence interval for the mean
difference between birth rates on weekdays and
weekends:
• We get our critical value tk* = 2.365 is calculated from a t
distribution with 7 degrees of freedom, so our 95%
confidence interval is:
• Since zero is not contained in this interval, we know the
difference is statistically significant!
6
Confidence Interval for Blackout Data
• We can calculate a 95% confidence interval for the mean
difference between birth rates on weekdays and
weekends:
• We get our critical value tk* = 2.365 is calculated from a t
distribution with 7 degrees of freedom, so our 95%
confidence interval is:
• Since zero is not contained in this interval, we know the
difference is statistically significant!
Two-Sample t test with unknown variances
• One more alternative: Suppose we are comparing two populations
that have different means but the same standard deviations.
• We want to infer about the difference between the means when the
standard deviation is unknown.
• We are assuming that both populations have the same standard
deviation but we have two estimates S12 and S 22 (the two samples
standard deviations).
• The best way to combine theses two estimates to give a more
informative estimator. The pooled estimator of the variance is
s 2p 
(n1  1)  s12  (n2  1)  s22
n1  n2  2
7
Two-Sample t test with unknown variances
The test statistics that should be used in this situation is
T
( x1  x2 )  0
1 1
s 2p   
 n1 n2 
Calculate the P-value by using the t distribution with ( n1  n2  2)
degrees of freedom and then compare it to the appropriate
significance level
Alternatively if we are testing a two-sided hypothesis we can
construct the appropriate CI:


 ( x  x )  t *  s 2  1  1  , ( x  x )  t *  s 2  1  1  
1
2
p
1
2
p



 n1 n2 
 n1 n2  

Matched Pairs
• Sometimes the two samples that are being compared are
matched pairs (not independent)
• Example: Sentences for crack versus powder cocaine
• We could test for the mean
difference between X1 = crack
sentences and X2 = powder
sentences
• However, we realize that these data
are paired: each row of sentences
have a matching quantity of
cocaine
• Our t-test for two independent
samples ignores this relationship
8
Matched Pairs Test
• First, calculate the difference d = X1 - X2 for each pair
• Then, calculate the mean and SD of the differences d
Sentences
Quantity
Crack
X1
Powder
X2
Difference
d = X 1 - X2
5
70.5
12
58.5
25
87.5
18
69.5
100
136
30
106.0
200
169.5
37
132.5
500
211.5
70.5
141.0
2000
264
87.5
176.5
5000
264
136
128.0
50000
264
211.5
52.5
150000
264
264
0.0
Matched Pairs Test
• Instead of a two-sample test for the difference between
X1 and X2, we do a one-sample test on the difference d
• Null hypothesis: mean difference between the two
samples is equal to zero
H0 : d= 0
versus Ha : d 0
• Usual test statistic when population SD is unknown:
• p-value calculated from t-distribution with df = 8
• P(T > 5.24) < 0.0005 so p-value < 0.001
• Difference between crack and powder sentences is
statistically significant at -level of 0.05
9
Matched Pairs Confidence Interval
• We can also construct a confidence interval for the mean
difference d of matched pairs
• We can just use the confidence intervals we learned for the onesample, unknown  case
• Example: 95% confidence interval for mean difference
between crack and powder sentences:
Summary of Two-Sample Tests
• Two independent samples with known 1 and 2
• We use two-sample Z-test with p-values calculated using the
standard normal distribution
• Two independent samples with unknown 1 and 2
• We use two-sample t-test with p-values calculated using the t
distribution with degrees of freedom equal to the smaller of n 1-1
and n2-1
• Two independent samples with unknown 1 and 2 and
assume they are equal
• We use two-sample t-test with pooled variance estimator. The pvalues is calculated using the t distribution with n1+n2-2 degrees of
freedom
• Two samples that are matched pairs
• We first calculate the differences for each pair, and then use our
usual one-sample t-test on these differences
10
Summary of Two-Sample Tests
• JMP!
• How to make a one sample t-test like we have learned
• How to make two sample t-tests
11