Admission in India

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6.6: Small-sample inference for a proportion

7.1: Large sample comparisons for two independent sample means.

7.2: Difference between two large sample proportions.

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So far, we have been making estimates and inferences about a single sample statistic

Now, we will begin making estimates and inferences for two sample statistics at once.

• many real-life problems involve such comparisons

• two-group problems often serve as a starting point for more involved statistics, as we shall see in this class.


Two independent random samples: • Two subsamples, each with a mean score for some other

variable• example: Comparisons of work hours by race or sex • example: Comparison of earnings by marital status

Two dependent random samples: • Two observations are being compared for each “unit” in

the sample• example: before-and-after measurements of the same

person at two time points • example: earnings before and after marriage• husband-wife differences


Hypothesis testing as we have done it so far:

Test statistic: z = (Ybar - o) / (s /SQRT(n))

What can we do when we make inferences about a difference between population means (2 - 1)?

• Treat one sample mean as if it were o ?

• (NO: too much type I error)

• Calculate a confidence interval for each sample mean and see if they overlap?

• (NO: too much type II error)5

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Is Y2 –Y1an appropriate way to evaluate 2 - 1?

• Answer: Yes. We can appropriately define (2 - 1) as a parameter of interest and estimate it in an unbiased way with (Y2 – Y1) just as we would estimate with Y.

• This line of argument may seem trivial, but it becomes important when we work with variance and standard deviations.


Comparing standard errors: A&F 213: formula without derivation

Is s2Ybar2 - s2

Ybar1an appropriate way to estimate 2

(Ybar2-Ybar1)?

• No!

2(Ybar2-Ybar1)= 2

(Ybar2) - 2(Ybar2,Ybar1) + 2(Ybar1)

• Where 2(Ybar2,Ybar1) reflects how much the observations for the two groups are dependent.

• For independent groups, 2(Ybar2,Ybar1) = 0, so 2

(Ybar2-Ybar1)= 2(Ybar2) + 2

(Ybar1)7


The parameter of interest is 2 - 1

Assumptions:

• the sample is drawn from a random sample of some sort,

• the parameter of interest is a variable with an interval scale,

• the sample size is large enough that the sampling distribution of Ybar2 – Ybar1 is approximately normal.

• The two samples are drawn independently


The null hypothesis will be that there is no difference between the population means. This means that any difference we observe is due to random chance.

Ho: 2 - 1 = 0

• (We can specify an alpha level now if we want)

Q: Would it matter if we used

Ho: 1 - 2 = 0 ?

Ho: 1 = 2 ?


The test statistic has a standard form:

• z = (estimate of parameter – Ho value of parameter) standard error of parameter

Q: If the null hypothesis is that the means are the same, why do we estimate two different standard deviations?

10

2

22

1

21

12 0)(

ns

ns

YYz


P-value of calculated z: • Table A

• Stata: display 2 * (1 – normal(z) )

• Stata: testi (no data, just parameters)

• Stata: ttest (if data file in memory)


Step 5: Conclusion.

Compare the p-value from step 4 to the alpha level in step 1.If p < α, reject H0 If p ≥ α, do not reject H0

State a conclusion about the statistical significance of the test.

Briefly discuss the substantive importance of your findings.


Do women spend more time on housework than men?

Data from the 1988 National Survey of Families and Households:• sex sample size mean hours s.d• men 4252 18.1 12.9• women 6764 32.6

18.2

The parameter of interest is 2 - 1


1. Assumptions: random sample, interval-scale variable, sample size large enough that the sampling distribution of 2 - 1is approximately normal, independent groups

2. Hypothesis: Ho: 2 - 1= 0

3. Test statistic: z = ((32.6 – 18.1) – 0) / SQRT((12.9)2/4252 + (18.2)2/6764) = 48.8

4. p-value: p<.001

5. conclusion:

a. reject H0: these sample differences are very unlikely to occur if men and women do the same number of hours of housework.

b. furthermore, the observed difference of 14.5 hours per week is a substantively important difference in the amount of housework.


housework example with 99% interval: c.i….

= (32.6 – 18.1) +/- 2.58*( √((12.9)2/4252 + (18.2)2/6764))

= 14.5 +/- 2.58*.30

= 14.5 +/- .8, or (13.7,15.3)

By this analysis, the 99% confidence interval for the difference in housework is 13.7 to 15.3 hours.

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2

22

1

21

12..n

s

n

szYYic


Immediate (no data, just parameters)• ttesti 4252 18.1 12.9 6764 32.6 18.2, unequal

• Q: why ttesti with large samples? For the immediate command, you need the following:

• sample size for group 1 (n = 4252)

• mean for group 1

• standard deviation for group 1

• sample size for group 2

• mean for group 2

• standard deviation for group 2

• instructions to not assume equal variance (, unequal)16


. ttesti 4252 18.1 12.9 6764 32.6 18.2, unequal

Two-sample t test with unequal variances

------------------------------------------------------------------------------ | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]---------+-------------------------------------------------------------------- x | 4252 18.1 .1978304 12.9 17.71215 18.48785 y | 6764 32.6 .221294 18.2 32.16619 33.03381---------+--------------------------------------------------------------------combined | 11016 27.00323 .1697512 17.8166 26.67049 27.33597---------+-------------------------------------------------------------------- diff | -14.5 .2968297 -15.08184 -13.91816------------------------------------------------------------------------------Satterthwaite's degrees of freedom: 10858.6

Ho: mean(x) - mean(y) = diff = 0

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 t = -48.8496 t = -48.8496 t = -48.8496 P < t = 0.0000 P > |t| = 0.0000 P > t = 1.0000


. ttest YEARSJOB, by(nonstandard) unequal

Two-sample t test with unequal variances------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]---------+-------------------------------------------------------------------- 0 | 980 9.430612 .2788544 8.729523 8.883391 9.977833 1 | 379 7.907652 .3880947 7.555398 7.144557 8.670747---------+--------------------------------------------------------------------combined | 1359 9.005887 .2290413 8.443521 8.556573 9.4552---------+-------------------------------------------------------------------- diff | 1.522961 .4778884 .5848756 2.461045------------------------------------------------------------------------------ diff = mean(0) - mean(1) t = 3.1869Ho: diff = 0 Satterthwaite's degrees of freedom = 787.963

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 0.9993 Pr(|T| > |t|) = 0.0015 Pr(T > t) = 0.0007


. ttest conrinc if wrkstat==1, by(wrkslf) unequal

Two-sample t test with unequal variances------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]---------+--------------------------------------------------------------------self-emp | 190 48514.62 2406.263 33168.05 43768.03 53261.2 someone | 1263 34417.11 636.9954 22638 33167.43 35666.8---------+--------------------------------------------------------------------combined | 1453 36260.56 648.5844 24722.9 34988.3 37532.82---------+-------------------------------------------------------------------- diff | 14097.5 2489.15 9191.402 19003.6------------------------------------------------------------------------------ diff = mean(self-emp) - mean(someone) t = 5.6636Ho: diff = 0 Satterthwaite's degrees of freedom = 216.259

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 1.0000 Pr(|T| > |t|) = 0.0000 Pr(T > t) = 0.0000


In 1982 and 1994, respondents in the General Social Survey were asked: “Do you agree or disagree with this statement? ‘Women should take care of running their homes and leave running the country up to men.’”

• Year Agree Disagree Total

• 1982 122 223 345

• 1994 268 1632 1900

• Total 390 1855 2245

Do a formal test to decide whether opinions differed in the two years.


The parameter of interest is π2 - π1

Assumptions:

• the sample is drawn from a random sample of some sort,

• the parameter of interest is a variable with an interval scale,

• the sample size is large enough that the sampling distribution of Pihat2 – Pihat1 is approximately normal.

• The two samples are drawn independently


The null hypothesis will be that there is no difference between the population proportions. This means that any difference we observe is due to random chance.

Ho: π2 - π1 = 0

(State an alpha here if you want to.)


The test statistic has a standard form: z = (estimate of parameter – Ho value of parameter)

standard error of parameter

Where pihat is the overall weighted average

• This means we are assuming equal variance in the two populations.

• Q: why do we use an assumption of equal variance to estimate the standard error for the t-test?

23

21

12

11ˆ1ˆ

)ˆˆ(

nn

z


P-value of calculated z: • Table A, or

• Stata: display 2 * (1 – normal(z) ), or

• Stata: testi (no data, just parameters)

• Stata: ttest (if data file in memory)


Conclusion:

Compare the p-value from step 4 to the alpha level in step 1.If p < α, reject H0 If p ≥ α, do not reject H0

State a conclusion about the statistical significance of the test.

Briefly discuss the substantive importance of your findings.


1. Assumptions: random sample, interval-scale variable, sample size large enough that the sampling distribution of 2 - 1is approximately normal, independent groups

2. Hypothesis: Ho: π2 - π1= 0

3. Test statistic:

z = (122/345 – 268/1900) /

SQRT[(390/2245)*(1 - 390/2245)*(1/345 + 1/1900)]

= 9.59

4. p-value: p<<.001

5. conclusion:

a. reject H0: attitudes were clearly different in 1994 than in 1982.

b. furthermore, the observed difference of .21 is a substantively important change in attitudes. 26


confidence interval:

Notice that there is no overall weighted average Pihat, as there is in a significance test for proportions.• Instead, we estimate two separate variances from the

separate proportions. • Why?

27

2

22

1

1112

)1()1(..

n

PP

n

PPzPPic


. prtesti 345 .3536 1900 .1411

STATA needs the following information:• sample size for group 1 (n = 345)• proportion for group 1 (p = 122/345)• sample size for group 2 (n = 1900)• proportion for group 2 (p = 268/1900)


. prtesti 345 .3536 1900 .1411

Two-sample test of proportion x: Number of obs = 345 y: Number of obs = 1900

------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- x | .3536 .0257393 .3031518 .4040482 y | .1411 .0079865 .1254467 .1567533-------------+---------------------------------------------------------------- diff | .2125 .0269499 .1596791 .2653209 | under Ho: .0221741 9.58 0.000------------------------------------------------------------------------------

Ho: proportion(x) - proportion(y) = diff = 0

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 z = 9.583 z = 9.583 z = 9.583 P < z = 1.0000 P > |z| = 0.0000 P > z = 0.0000

Note the use of one standard error (unequal variance) for the confidence interval, and another (equal variance) for the significance test.


. prtest nonstandard if (RACECEN1==1 | RACECEN1==2), by(RACECEN1)

Two-sample test of proportion 1: Number of obs = 1389 2: Number of obs = 260------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- 1 | .2800576 .0120482 .2564436 .3036716 2 | .3538462 .0296544 .2957247 .4119676-------------+---------------------------------------------------------------- diff | -.0737886 .0320084 -.1365239 -.0110532 | under Ho: .0307147 -2.40 0.016------------------------------------------------------------------------------ diff = prop(1) - prop(2) z = -2.4024 Ho: diff = 0

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(Z < z) = 0.0081 Pr(|Z| < |z|) = 0.0163 Pr(Z > z) = 0.9919


. gen byte wrkslf0=wrkslf-1(152 missing values generated)

. prtest wrkslf0 if wrkstat==1, by(sex)

Two-sample test of proportion male: Number of obs = 874 female: Number of obs = 743------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- male | .8272311 .0127876 .8021678 .8522944 female | .9044415 .0107853 .8833027 .9255802-------------+---------------------------------------------------------------- diff | -.0772103 .0167286 -.1099978 -.0444229 | under Ho: .0171735 -4.50 0.000------------------------------------------------------------------------------ diff = prop(male) - prop(female) z = -4.4959 Ho: diff = 0

Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(Z < z) = 0.0000 Pr(|Z| < |z|) = 0.0000 Pr(Z > z) = 1.0000


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