7.1
Inference Section for the mean of a population
Change: Population standard deviation () is now unknown The t distribution One-sample t confidence interval One-sample t test Matched pairs t procedures Robustness of t procedures
The t distribution:
The goal is to estimate or test for an unknown in the situation when is also unknown. Solution: estimate by s and use it intelligently in the formulas. Challenge: the distribution of the test statistic will change and will no longer be the standard normal distribution.
Sampling Normal Population, Unknown Standard Deviation
Suppose an SRS X1, ..., Xn is selected from a normally distributed population with mean and standard deviation . Assume that and are both unknown. We know that X ~ N , n When is unknown, we estimate its value with the sample standard deviation s.
(
)
Sampling Normal Population, Unknown Standard Deviation
The standard deviation of X can be estimated by
s SE X = n This quantity is called the standard error of the
sample mean. The test statistic (appropriately standardized sample mean) will no longer be normally distributed when we use the standard error. The test statistic will have a new distribution, called the t (or Student's t) distribution.
The t-test Statistic and Distribution
Suppose that an SRS of size n is drawn from an N(, ) population. Then the one-sample t-statistic
t=
x - 0 s n
has the t distribution with n 1 degrees of freedom. There is a different t distribution for each sample size. The degrees of freedom for the t-statistic "come" from the sample standard deviation s. The density curve of a t distribution with k degrees of freedom is symmetric about 0 and bell-shaped.
The t-test Statistic and Distribution
The higher the degrees of freedom (df) are, the narrower the spread of the t distribution
n1 < n2 df = n2 df = n1
0
As the df increase, the t density curve approaches the N(0, 1) curve more closely Generally it is more spread than the normal, especially if the df are small
One-sample t Confidence Interval
Suppose a SRS of size n is drawn from a population having unknown mean . A level C confidence interval for is
x t
*
s , n
or
* s * s x , x +t -t n n
Here t* is the value for the t density curve with the df = n-1. The area between t* and t* is C The interval is exact for the normal population and approximately correct for large n in other cases Note the standard error in the formula
Example 1
A mutual fund is trying to estimate the return on investment in companies that won quality awards last year. A random sample of 20 such companies is selected, and the return on investment is calculated. The mean of the sample is 14.75 and the standard deviation of the sample is 8.18. Construct a 95% confidence interval for the mean return on investment.
Example 2
From a running production of corn soy blend we take a sample to measure content of vitamin C. The results are: 26 31 23 22 11 22 14 31 Find a 95% confidence interval for the content of vitamin C in this production. Give the margin of error.
One-Sample t Test
Suppose that an SRS of size n is drawn from a population having unknown mean . To test the hypothesis H0: = 0 based on an SRS of size n, compute the one-sample t statistic
x - 0 t= s n
Note the standard error in the denominator.
One-Sample t Test
In terms of a random variable T with t (n1) distribution, the P-value for a test of H0: = 0 against... Ha: > 0 is P(T t) Ha: < 0 is P(T t) Ha: 0 is 2P(T | t |)
P-values
These P-values are exact if the population distribution is normal and are approximately correct for large n in other cases. Use Table D to know the range of the Pvalues, software gives exact p-values
Example 2 (vitamin C continued):
Test whether the vitamin C content from the sample conforms to specifications of 40.
Example (vitamin C continued):
Test whether the vitamin C content from the sample is lower than the specifications of 40.
Matched Pairs t Procedures
Inference about a parameter of a single distribution is less common than comparative inference. In certain circumstances a comparative study makes use of single-sample t procedures. In a matched pairs study, subjects are matched in pairs; the outcomes are compared within each matched pair. One typical situation here is "before" and "after" (quantitative) observations of the same subject.
Matched Pairs t Test
A matched pairs analysis is appropriate when there are two measurements or observations per each individual and we examine the change from the first to the second. Typically, the observations are "before" and "after" measures in some sense. For each individual, subtract the "before" measure from the "after" measure. Analyze the difference using the one-sample confidence interval or significance-testing t procedures (with H0: diff = 0).
Example
20 French teachers attend a course to improve their skills. The teachers take a Modern Language Association's listening test at the beginning of the course and at it's end. The maximum possible score on the test is 36. The differences in each participant's "before" and "after" scores have sample mean 2.5 and sample standard deviation 2.893. Is the improvement significant? Construct a 95% confidence interval for the mean improvement (in the entire population).
Robustness of t Procedures
A statistical inference is called robust if it's outcome is not sensitive to the violation of the assumptions made. Real populations are never exactly normal. Usefulness of the t procedures in practice depends on how strongly they are affected by non-normality. If they are not strongly affected, we say that they are robust. The t procedures are robust against non-normality of the population except in the case of outliers or strong skewness.
Robustness of t Procedures
Practical guidelines for inference on a single mean: Sample size < 15: Use t procedures if the are data close to normal; otherwise, don't. Sample size 15: Use t procedures except in the presence of outliers or strong skewness. Large sample size ( 40): Use t procedures even for clearly skewed distributions. Use normal quantile plot, histogram, stemplot, and/or boxplot to investigate these properties of data.
Assumption that data are SRS always important.
Using SAS
One
sample t
Proc ttest--gives confidence interval and significance test
For
matched pairs
Still use similar code as one sample t Or use the paired command
Section 7.2
Comparing Two Means
Topics
The two-sample z statistics The two-sample t procedures:
significance test confidence interval
Robustness of the two-sample procedures Small samples
Two-Sample Problems
Two-sample problems typically arise from a randomized comparative experiment with two treatment groups. (Experimental study) Comparing random samples separately selected from two populations is also a two-sample problem. (Observational study) Unlike matched pairs design, there is no matching of the units in the two sample, and the two samples may be of different sizes.
Notation for Two-Sample Settings
Population Population mean 1 2 Population standard deviation 1 2
1
2
Notation for Two-Sample Settings
Suppose an SRS of size n1 is selected from the 1st population, and another SRS of size n2 is selected from the 2nd population. Population Sample size 1 n1 2 n2 Sample mean
x1 x2
Sample standard deviation s1 s2
The Two-Sample z Statistic
A natural estimator of the difference 1 2 is the difference between the sample means: x1 - x2 From the rules of adding means and variances:
(population) mean of differences: 1 - 2
2 12 2 (population) variance of differences of sample standard deviations: + n1 n2
This expresses the mean and variance of the distribution of differences (of sample means) in terms of the parameters of the two original populations. If the two population distributions are both normal, then the distribution of the difference D is also normal.
Distribution of Two-Sample z Statistic
Suppose that x1 is the mean of an SRS of size n1 drawn from an N(1, 1) population and that x2 is the mean of an independent SRS of size n2 drawn from an N(2, 2) population. Then the two-sample z statistic is ( x - x ) - ( - )
z=
1
2
1
2
+ n1 n2
2 1
2 2
and it has the standard normal distribution, N(0, 1), as
Two Populations, Known Population Standard Deviations
If 1 and 2 are unknown, then a level C confidence interval for 1 2 is
x1 - x2 z
*
+ n1 n2
2 1
2 2
where P(-z* Z z*) = C.
Two Populations, Known Population Standard Deviations
We want to test H0: 1 = 2 against one of the following alternative hypotheses: Ha: 1 > 2
Ha: 1 < 2 Ha: 1 2
The z test statistic when known population SDs:
( x1 - x2 ) - ( 1 - 2 ) z=
2 12 2 + n1 n2
Inference Two Populations, Known Population Standard Deviations
Alternative Hypothesis Ha: 1 > 2 Ha: 1 < 2 Ha: 1 2 P-value P(Z > z) P(Z < z) 2*P(Z > | z |)
Two Populations, Unknown Population Standard Deviations
Suppose 1, 2, 1, and 2 are unknown Two-sample t statistic for difference in means:
t=
( x1 - x2 ) - ( 1 - 2 )
2 s12 s2 + n1 n2
is approximately t distributed with the df either approximated by software or taken as: min(n1 1, n2 1).
Two Populations, Unknown Population Standard Deviations
If 1, 2, 1, and 2 are unknown, then a level C confidence interval for 1 2 is
( x1 - x2 ) t
* P (tdf
* df
s s + n1 n2
* tdf ) = C
2 1
2 2
where
t
(from Table D). So, this is t. Degrees of freedom as before: min(n 1, n 1).
Example (metabolism rates for men and women):
Obs Gender Mass Rate Obs 1 2 3 4 5 6 7 8 9 10 11 12 M M F F F F M F F M F F 62 62.9 36.1 54.6 48.5 42 47.4 50.6 42 48.7 40.3 33.1 1792 1666 995 1425 1396 1418 1362 1502 1256 1614 1189 913 1 2 3 4 5 6 7 8 9 10 11 F F F F F F F F F F F 36.1 54.6 48.5 42 50.6 42 40.3 33.1 42.4 34.5 51.1 995 1425 1396 1418 1502 1256 1189 913 1124 1052 1347 Gender Mass Rate
Example (cont)
A back-to-back stem plot or boxplot is always a good idea:
1 8 0 0 1 6 0 0 R a t 1 4 0 0 e 1 2 0 0 1 0 0 0 F M G e n d e r
Example (Metabolism rate: women vs.
men)
Find a 95% CI for the difference in mean metabolism rates between men and women.
n1 = 12 n2 = 7 x1 = 1235.1 x2 = 1600 s1 = 188.3 s 2 = 189.2
Two Populations, Unknown Population Standard Deviations
Goal: test H0: 1 = 2 against one of the following alternative hypotheses when 1, 2 are unknown: Ha: 1 > 2 Ha: 1 < 2 x1 - x2 - 1 - 2 Ha: 1 2 t= 2 2 The t test statistic is
(
) (
)
s1 s2 + n1 n2
Degrees of freedom as before: min(n1 1, n2 1) or from
software. What does the "1
2" in the formula equal?
Two Populations, Unknown Population Standard Deviations 1, 2
Alternative Hypothesis Ha: 1 > 2 Ha: 1 < 2 Ha: 1 2 P-value P(T t) P(T t) 2*P(T | t |)
Example: Do women have a different metabolism rate than men?
Robustness of Two-Sample t Test
Even more robust than one-sample t methods
robust against non-normal population distributions, in particular if the population distributions are symmetric and the two sample sizes are equal.
Outliers are always a problem may need to be eliminated. Skewness is less important for not-toosmall sample sizes. t procedures are rather conservative, so your calculated P-values may be even larger than the "real" P-values. This is good (safe).
df for two-sample t procedures
df as before: min(n1 1, n2 1) or from software The choice of min(n1 1, n2 1) is conservative. Software will usually give smaller P-values.
More correct.
In our example with metabolism rates, software will give df = 12.6 (12 in Table D)
Here no difference in final conclusion...
The pooled two-sample t-test
If the variances of the two groups could be assumed to be equal, then a pooled twosample t test can be used (pages 461465).
Using SAS
Two
sample t
Still use similar code as one sample t Add class command to designate two groups or populations