91Ó°ÊÓ

In hypothesis testing, one important concept is sample proportions. A sample proportion represents the ratio of individuals in a sample with a particular attribute to the total number of individuals in that sample. In our exercise, we calculate the sample proportions for two different groups on the common attribute they possess. For the first sample of 20 people, 10 have the attribute, thus the sample proportion is calculated as:
Step 1: Determine sample proportion for first group: The first sample has 10 people with the attribute out of 20, making the sample proportion: For the second sample, which consists of 2000 people where 1404 have the attribute, the sample proportion is: Step 1: Determine sample proportion for second group: We compare the sample proportions to understand differences between the two independent samples. The formulas we use are:

• For the first sample:
  
  
  p_1 = \(\frac{10}{20} = 0.5\)
  
• For the second sample: p_2 = \(\frac{1404}{2000} = 0.702\)
  

Confidence intervals help us estimate the range within which the true difference between two population proportions lies, based on our sample data. It provides a range of values which is believed to contain the true parameter with a specified level of confidence, such as 95%. In our example, we calculate a 95% confidence interval for the difference between the two proportions:



A 95% confidence level means we are 95% confident the true difference lies within this interval.

Here’s how we calculate:

• Difference in proportions: \(p_1 - p_2\) = -0.202
  
  
• Standard error formula includes both sampling variations:
• Confidence interval is calculated as:
  
  
  
  
  
  
  
• 95% Confidence Interval = \(-0.202\pm 1.96\times 0.11\approx [-0.418, 0.014]\)
  
  
  
  

The pooled proportion is an estimate of a single proportion based on the combined data from both samples. This concept is essential in hypothesis testing when comparing two proportions. By pooling the data, we get a weighted average of the two sample proportions.



Calculation involves summing the individual successes and the total sample size:

• Combined successes: \(x_1+x_2 = 10+1404 = 1414\)
• Total sample size: \(n_1+n_2 = 20+2000 = 2020\)
  
  
  
  
  
  
  
  Pooled Proportion = \(\hat{p}=\frac{1414}{2020}=0.7\)
  

The test statistic helps determine whether to reject the null hypothesis. This value is derived from a specific formula and compared against critical values to make a decision. For our scenario, we calculate the test statistic by:

• Using the difference of sample proportions: \(p_1-p_2\)
  
  [...]
  
• Pooled proportion \(\hat{p}\)
  
• Standard error formula which incorporates the pooled proportion and sample sizes
  [...]
  
  
  
  
  The Test Statistic (z value) = \(\frac{0.5-0.702}{\sqrt{{0.7(1-0.7)(\frac{1}{20}+\frac{1}{2000})}}}\approx-2.008\)

In hypothesis testing, the null hypothesis \((H_0)\) is a statement we aim to test. It represents no effect or no difference and is assumed true until proven otherwise. In our problem, we tested: \(H_0:p_1=p_2\)




The goal is to examine if the difference between sample proportions is due to random chance or a real difference:

• If the test statistic falls within critical values for a chosen significance level \((\alpha=0.05)\), we reject \(H_0\).
  
  
• Critical values for two-tailed test: \(\pm1.96\). Since our calculated z is -2.008, we reject \(H_0\), indicating significant difference between proportions.