Problem Description:
This inferential statistics homework focuses on the testing of proportions and means to make inferences about populations. Two key questions are addressed in this homework: (1) Is the proportion of schools with Grade B different from 50%? (2) Is the population mean of reading scores greater than 500?
Solution:
Test of Proportion and Mean
1. The 95% confidence interval of proportion for Grade B was obtained to test the following hypothesis
Null Hypothesis: The probability that a school will have a Grade B is 0.5 (p = 0.5)
Alternative Hypothesis: The probability that a school will have a Grade B is not equal to 0.5
The confidence interval was obtained in SPSS and the result is displayed below;
Confidence Interval Type 
Parameter  Estimate  95% Confidence Interval  

Lower  Upper  
One Sample Binomial Success rate(ClopperPearson) 
Probability (Grade = B) 
.220  .115  .360 
The result indicates that the proportion of all schools with grade B can be as low as 11.5% and can be as high as 36%.
2. To test whether the true proportion of schools with grade B is less than 0.5, a test of proportion was carried out in SPSS with the following hypothesis and rejection rule
Null Hypothesis: The probability that a school will have a Grade B is 0.5 (p = 0.5)
Alternative Hypothesis: The probability that a school will have Grade B is less than 0.5 (p<0.5)
Rejection rule: We reject the null hypothesis if the pvalue is less than 0.05
The test result has a pvalue less than 0.05, we reject the null hypothesis and conclude that, the probability that a school has a grade B is less than 0.05.
Total N  50 
Test Statistics  11.000 
Standard Error  3.536 
Standardized Test Statistics  3.818 
Asymptotic Sig. (2sided test)  .000 
3. The confidence interval for the population mean is given in the table below
Descriptives
Descriptives 



Statistic 
Std. Error 

Math Score 
Mean 
537.82 
6.758 

95% Confidence Interval for Mean 
Lower Bound 
524.25 


Upper Bound 
551.40 


5% Trimmed Mean 
538.36 


Median 
525.00 


Variance 
2329.228 


Std. Deviation 
48.262 


Minimum 
438 


Maximum 
620 


Range 
182 


Interquartile Range 
83 


Skewness 
.115 
.333 

Kurtosis 
1.012 
.656 
The 95% confidence interval of the population mean (524.25, 551.40) indicates that we 95% confident that the population average math score is between 524.25 and 551.40
4. To test whether the true population mean of reading score is greater than 500, the following hypothesis was tested at 0.05 level of significant
Null hypothesis: The population mean reading score is 500
Alternative hypothesis: The population mean reading score is less than 500
Rejection rule: Reject the null hypothesis if the pvalue is less than 0.05
The test result is displayed below;
OneSample Statistics 



N 
Mean 
Std. Deviation 
Std. Error Mean 
Reading Score 
51 
534.67 
47.903 
6.708 
OneSample Statistics 



N 
Mean 
Std. Deviation 
Std. Error Mean 
Reading Score 
51 
534.67 
47.903 
6.708 
The test result has a pvalue greater than 0.05, the null hypothesis is not rejected and hence we conclude that the population mean reading score is 500. [t (50) = 5.168, p>0.05]