## 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.