# Implementing Inferential Statistics in SPSS to Test Proportions and Means

A comprehensive SPSS homework on inferential statistics, where we delve into the realms of proportions and means to make insightful inferences about various populations. In this homework, we address two pivotal questions: Is the proportion of schools with Grade B different from 50%, and is the population mean of reading scores greater than 500? Let's explore the statistical journey that leads to these answers.

## 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(Clopper-Pearson)
Probability
.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 p-value is less than 0.05

The test result has a p-value 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 Standard Error 3.536 Standardized Test Statistics -3.818 Asymptotic Sig. (2-sided test) 0

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 p-value is less than 0.05

The test result is displayed below;

One-Sample Statistics

N

Mean

Std. Deviation

Std. Error Mean

51

534.67

47.903

6.708

One-Sample Statistics

N

Mean

Std. Deviation

Std. Error Mean