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I have a data set of 412 schools and the mean SAT scores for these schools in 2012. I have created a dummy variable called high-income and low-income school based on the number of students that receive free or reduced lunches.

I want to conduct a Hypothesis tests for the difference of two means where I hope to set up the following hypothesis

H0-There is no average difference in SAT score between low-income and high income schools

HA- There is a difference in mean SAT scores between the low-income and high income schools.

I'm not sure if I have fulfilled the assumptions to carry out a difference of two means t-test

There are 1700 schools in NYC, and the dataset I have has 412 schools. However I am not sure if this data comes from an independent random samples. Also within these 412 schools only 142 schools fall as "high-poverty" and "low-poverty" category.

see blow for the histograms for both groups enter image description here enter image description here

In such a situation should I not conduct the hypothesis test I have proposed

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    $\begingroup$ Unless you have reason to suspect that some particular biased non-random method accounts for your having data for only about 400 schools in 1700, it seems worthwhile looking to see if there is a difference btw the two gps. In writing up results, admit you have data for only about 1/4 of the NYC schools, and give any information you may have about selection of those schools. // The SAT scores for the various schools are likely nearly normal. If you have about 200 schools in each group, it seems OK to do a Welch 2-sample t test. // To say for sure, it would help to see histograms of the 2 gps. $\endgroup$ – BruceET Mar 22 '20 at 21:16
  • $\begingroup$ @BruceET added the histograms $\endgroup$ – Nathan123 Mar 22 '20 at 21:34
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    $\begingroup$ Looks OK. If it were my project, then I'd try to find out why there's a gap in the bottom histogram (bin centered at 525. In a formal presentation, you'd want to plot the histograms so they have the same horizontal scales--for easier visual comparison. $\endgroup$ – BruceET Mar 22 '20 at 22:24
  • $\begingroup$ Upon closer inspection of your histograms I see only about 125+12=137 schools, not 412 as in your question, and not approximately equal numbers with high and low poverty as speculated in my first comment. With such an imbalance in the two sample sizes, and unequal variances, it is not obvious whether you will find a difference with data as in your histograms, I'm trying something similar in Answer format. – $\endgroup$ – BruceET Mar 23 '20 at 8:55
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With fake data (generated in R) that I hope roughly matches the data you used to make your histograms, I did a Welch 2 sample t-test, and found a significant difference.

Data summary:

summary(hi.pov)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  302.1   357.0   376.0   381.4   397.7   575.0 
sd(hi.pov);  length(hi.pov)
[1] 39.51679
[1] 126
summary(lo.pov)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  465.6   545.6   586.5   576.7   615.0   680.0 
sd(lo.pov);  length(lo.pov)
[1] 70.99293
[1] 12

Welch t test: highly significant

t.test(hi.pov,lo.pov)

        Welch Two Sample t-test

data:  hi.pov and lo.pov
t = -9.3894, df = 11.658, p-value = 8.834e-07
alternative hypothesis: 
   true difference in means is not equal to 0
95 percent confidence interval:
 -240.6984 -149.7898
sample estimates:
mean of x mean of y 
 381.4178  576.6619 

Histograms of fake data:

par(mfrow=c(2,1))
 hist(hi.pov, xlim=c(300,680), col="skyblue2")
 hist(lo.pov, xlim=c(300,690), col="skyblue2")
par(mfrow=c(1,1))

enter image description here

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