Problem Description:
The hypothesis testing using the R assignment involves applying the six classical steps in hypothesis testing to two different scenarios. The first scenario deals with testing the fairness of a coin through a proportion test, while the second involves a chi-square test of association to explore potential relationships between categorical variables.
Example 1: Testing Coin Fairness
H0: p = 0.5 (The coin is fair)
H1: p ≠ 0.5 (The coin is not fair)
Note: this is a two-tailed test since the alternative hypothesis carries ≠ a sign. It can either be greater than 0.5 or less than 0.5
Formulate the Hypothesis:
- Null Hypothesis (H0): =0.5p=0.5 (The coin is fair)
- Alternative Hypothesis (H1): ≠0.5p=0.5 (The coin is not fair)
Specify the Level of Significance:
=0.05α=0.05 (5%)
State the Test Statistic:
- Use Z-test since conditions
Z=(sample statistic-null parameter)/(standard error)
State the Decision Rule:
- Reject the null hypothesis if the p-value is less than 0.05.
Computation of Test Statistic and P-value:
- Sample proportion =0.46p=0.46
- Standard Error 0.07SE(P)=0.07
- Test Statistic −0.571Z=−0.571
- P-value =2⋅(1−0.7157)=0.568=2⋅(1−0.7157)=0.568
Comparison/Decision:
- Since the p-value (0.568) is greater than 0.05, we fail to reject the null hypothesis, concluding that the coin is fair.
95% Confidence Interval:
- ±1.96)P±1.96⋅SE(P)
- Confidence Interval: [0.323,0.597][0.323,0.597]
Interpretation: We are 95% confident that the true proportion of success in a coin toss experiment lies between 0.323 and 0.597.
Example 2: Chi-square Test of Association
Formulate the Hypothesis:
- Null Hypothesis (H0): There is no association between the row and column variables.
Specify the Level of Significance:
- 0.05α=0.05 (5%)
State the Test Statistic:
- 2=∑(2χ2=∑E(O−E)2, df=(−1)⋅(−1)df=(r−1)⋅(c−1)
State the Decision Rule:
- Reject the null hypothesis if the p-value is less than 0.05.
Computation of Test Statistic and P-value:
- Compute 2χ2 and find the p-value using a chi-square table or statistical software.
Comparison/Decision:
- Compare the p-value with 0.05 and conclude.
Share and Discuss Sample Size and Variation:
- Bigger Sample: Increase in test statistic, decrease in p-value.
- Smaller Sample: Decrease in test statistic, increase in p-value.
- Similar Group Values: Small test statistic, high p-value.
- Different Group Values: Large test statistic, low p-value.