Question 3: Is there a significant difference between English Averages and Mathematics Averages for these schools?
1. Hypothesis
H0= There is not a significant difference between the English Averages and Math Averages for these schools.
HA≠There is a significant difference between the English Averages and Math Averages for these schools.
or
H0:μEμM =0
HA:μEμM≠0
2.The conditions
When creating this sample I created it randomly through the use of stat crunch. This concludes that this is a new random sample meaning the first condition is met. I then conducted a twosample T hypothesis test with a QQ plot. The quartile plot was roughly straight. Since the conditions are met I know that the sample is nearly normal.
3.Models & Procedures
The sampling model is tdf which is t40.92 . The procedure used is a twosample T hypothesis test. To obtain this sample in stat crunch first you must select stat>T stat>two sample>with data. Then one the box opens for sample 1 values in, select sample(ENGLISH AVERAGE). Then under sample 2 values in, select sample(MATH AVERAGE). Then for optional graphs select QQ plot then calculate.
4.The mechanics
The first step in finding the summary stats on stat crunch is by going to the new sample you creating and clicking on stat>summary stat> column. Then when the box opens under Select column(s): select sample(ENGLISH AVERAGE) and sample(MATH AVERAGE). Then under statistics selects n, mean, and standard deviation. The summary stats for sample(ENGLISH AVERAGE) are n=24, mean=20.291667, standard deviation=4.1856813. The summary stats for sample(MATH AVERAGE) are n=25, mean=22.52, standard deviation=2.9171904. Then in order to find df, use the two sample T hypothesis test constructed earlier. This test provides df, sample difference, sample error, t stat, and p value. DF:40.923461, sample difference:2.2283333, sample error:1.0345999, t stat or correct notation:2.1538116, p value: 0.0372. The t score is 2.15 and the p value is 0.037.
5. The conclusion
Since the P value is very small(0.037) we reject the null hypothesis. There is strong evidence that there is a difference in means between the English average and Math average.
Question 4: Is there statistical evidence that the English Averages and Mathematics Averages are different within each school?
1. Hypothesis
H0= There is no statistical evidence of different scores between each school.
HA≠There is statistical evidence of different scores between each school.
or
H0:μdiff =0
HA:μdiff≠0
2.The conditions
When creating this sample I created it randomly through the use of stat crunch. This concludes that this is a new random sample meaning the first condition is met. I then created a difference histogram using a paired T hypothesis test through stat crunch. The histogram is unimodal and roughly symmetric. The data are matched. All of the conditions are met.
3. Models and Procedures
The sampling model is t141. The procedure that needs to be performed is a paired t hypothesis test.
4. The Mechanics
In order to calculate the summary stats when I created the paired t hypothesis test, I selected save the differences. Due to this selection, a new column was created named differences. I then based my summary stats off of the differences. I clicked on stat>summary stat> column. Then when the box opens under Select column(s): select differences. Then under statistics select n, mean, and standard deviation. The summary stats then open up in a separate box n:14, mean:1.7857143, standard deviation: 5.2795479. Next, I created a paired t hypothesis test. I created this by selecting stat>t stat>paired. Then when the box opened up under sample 1 in: I selected sample(ENGLISH AVERAGE) and under sample 2 in: I selected sample( MATH AVERAGE) then selected compute. The mean, standard error, df, t stat, and p value are calculated. the mean is 1.7857143, standard error:1.4110185, df: 13, t stat or correct notation: 1.2655498, and p value:0.2279. The t score is 1.27 and the p value is 0.228.
5. The Conclusion
I do not reject the null hypothesis since the p value(0.228) is large. There is not sufficient enough evidence that the averages are different within each school.
Summary statistics:

Two sample T hypothesis test:μ_{1} : Mean of Sample(ENGLISH AVERAGE) μ_{2} : Mean of Sample(MATHEMATICS AVERAGE) μ_{1}  μ_{2} : Difference between two means H_{0} : μ_{1}  μ_{2} = 0 H_{A} : μ_{1}  μ_{2} ≠ 0 (without pooled variances) Hypothesis test results:

Summary statistics:

Paired T hypothesis test:μ_{D} = μ_{1}  μ_{2} : Mean of the difference between Sample(ENGLISH AVERAGE) and Sample(MATHEMATICS AVERAGE) H_{0} : μ_{D} = 0 H_{A} : μ_{D} ≠ 0 Hypothesis test results:

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