“Ah, the elusive nature of truth. In this worldly existence, we often find ourselves on the precipice between doubt and certainty. What tools, then, might we employ to tread carefully along this precipice? Let us humbly turn our gaze towards the realm of hypothesis tests, for it offers us a structured pathway to discern the likely from the unlikely, granting a modicum of clarity in a world abounding with questions.”

Chi-Squared Test for Independence

The Chi-Squared Test for Independence allows us to understand the relationship between two categorical variables. It helps us answer questions about the independence of data sets.

#### 1. Check Assumptions

1. Normality
2. Each cell in the contingency table should have an expected frequency of 5 or more.

3. Independence
4. The sample size should not exceed 10% of the population, as per the 10% rule, to ensure independence.

5. Random Sampling
6. Data should be randomly sampled from the population.

#### 2. State Hypotheses

Null Hypothesis: $$H_0:$$ Variables are independent (no relation)

Alternative Hypothesis: $$H_a:$$ Variables are not independent (relation)

#### 3. Conclude Test

If $$p < \alpha$$, reject $$H_0$$.

#### Instructions for TI-84

Go to STAT > TESTS > C:$$\chi^2$$-Test.

Two-Sample t-test for Population Mean

The Two-Sample t-test for Population Mean is useful when comparing the means of two separate groups. It helps to identify if the difference in means is statistically significant.

#### 1. Check Assumptions

1. Normality
2. If either sample size is smaller than 30, graphical methods such as dot plots or histograms should be used to check for normality. If both sample sizes are large $$( n\geq 30$$), the CLT can be assumed to hold.

3. Independence
4. The sample size should not exceed 10% of the population, as per the 10% rule, to ensure independence.

5. Random Sampling
6. Data should be randomly sampled from the population.

#### 2. State Hypotheses

Null Hypothesis: $$H_0: \mu_1 = \mu_2$$

Alternative Hypothesis: $$H_a: \mu_1 \neq \mu_2$$

#### 3. Conclude Test

If $$p < \alpha$$, reject $$H_0$$.

#### Instructions for TI-84

Go to STAT > TESTS > 2-SampTTest.

One-Sample t-test for Population Mean

The One-Sample t-test for Population Mean evaluates how a sample mean compares to a known or hypothesized population mean.

#### 1. Check Assumptions

1. Normality
2. If the sample size is large ($$n \geq 30$$), the Central Limit Theorem can be assumed to hold. For smaller sample sizes, make a dot plot or histogram and check the shape of the distribution.

3. Independence
4. The sample size should not exceed 10% of the population, as per the 10% rule, to ensure independence.

5. Random Sampling
6. Data should be randomly sampled from the population.

#### 2. State Hypotheses

Null Hypothesis: $$H_0: \mu = \mu_0$$, where $$\mu_0$$ is our assumed population mean.

Alternative Hypothesis: $$H_a: \mu \neq \mu_0$$

#### 3. Conclude Test

If $$p < \alpha$$, reject $$H_0$$.

#### Instructions for TI-84

Go to STAT > TESTS > 2:T-Test.

Two-Sample Z-test for Population Proportion

This test helps us to compare the proportions of successes in two independent samples, usually to determine if different groups are performing differently.

#### 1. Check Assumptions

1. Normality
2. Both $$n_1p_1$$ and $$n_1(1-p_1)$$, and $$n_2p_2$$ and $$n_2(1-p_2)$$ should be greater than or equal to 10.

3. Independence
4. The sample size should not exceed 10% of the population, as per the 10% rule, to ensure independence.

5. Random Sampling
6. Data should be randomly sampled from the population.

#### 2. State Hypotheses

Null Hypothesis: $$H_0: p_1 = p_2$$

Alternative Hypothesis: $$H_a: p_1 \neq p_2$$

#### 3. Conclude Test

If $$p < \alpha$$, reject $$H_0$$.

#### Instructions for TI-84

Go to STAT > TESTS > 6:2-PropZTest.

One-Sample Z-test for Population Proportion

This test helps us evaluate if a sample proportion is significantly different from a hypothesized population proportion.

#### 1. Check Assumptions

1. Normality
2. $$np$$ and $$n(1-p)$$ should be greater than or equal to 10.

3. Independence
4. The sample size should not exceed 10% of the population, as per the 10% rule, to ensure independence.

5. Random Sampling
6. Data should be randomly sampled from the population.

#### 2. State Hypotheses

Null Hypothesis: $$H_0: p = p_0$$

Alternative Hypothesis: $$H_a: p \neq p_0$$

#### 3. Conclude Test

If $$p < \alpha$$, reject $$H_0$$.

#### Instructions for TI-84

Go to STAT > TESTS > 5:1-PropZTest.

## Exercises

When you carry out a hypothesis test make sure you follow the procedure laid-out above for each test (check assumptions, state hypotheses, etc.).

1. Perform a Chi-Squared Test for Independence on the following table, which shows the relationship between gender and career choice. Each cell has a count greater than 5, and the sample is less than 10% of the population. What do you conclude?
2. Career Male Female
Engineer 30 20
Doctor 25 25
3. Using the Two-Sample t-test for Population Mean, compare the test scores of two different classes. Class A: [85, 90, 88, 76, 92, 87, 89, 81, 86, 88, 91, 84, 85, 90, 88, 76, 92, 87, 89, 81, 86, 88, 91, 84, 85, 90, 88, 76, 92, 87, 89], Class B: [78, 81, 86, 74, 80, 79, 82, 85, 77, 76, 81, 80, 82, 83, 78, 76, 79, 81, 85, 77, 76, 80, 82, 83, 78, 76, 79, 81, 85, 77, 76]. Is there a significant difference?
4. Apply the One-Sample t-test for Population Mean to determine whether a basketball team’s average points per game [85, 90, 76, 92, 88, 85, 90, 76, 92, 88, 85, 90, 76, 92, 88, 85, 90, 76, 92, 88, 85, 90, 76, 92, 88, 85, 90, 76, 92, 88] significantly differ from the league average of 80.
5. Conduct a Two-Sample Z-test for Population Proportion to investigate whether Brand A (35 out of 100 customers satisfied) and Brand B (25 out of 80 customers satisfied) have the same customer satisfaction rate.
6. Utilize the One-Sample Z-test for Population Proportion to see if 30 out of 100 students in your school prefer coffee to tea, and compare it to the national average preference rate of 0.35.
7. Given the p-value of 0.02 for a Chi-Squared Test for Independence, does this lead you to reject the null hypothesis at alpha = 0.05?
8. Using a Two-Sample t-test for Population Mean, test whether there is a significant difference in the average heights of men and women in two samples of 35 individuals each. Men: [175, 180, 169, 172, 177, 178, 176, 179, 170, 173, 174, 181, 168, 175, 176, 179, 171, 180, 174, 177, 182, 173, 171, 178, 176, 177, 169, 172, 175, 181, 178, 170, 174, 179, 171], Women: [160, 165, 157, 163, 161, 164, 159, 158, 162, 166, 161, 160, 165, 163, 159, 162, 164, 160, 157, 166, 163, 161, 164, 158, 162, 160, 165, 157, 163, 164, 160, 162, 159, 161, 166].
9. Conduct a One-Sample t-test for Population Mean on exam scores from a class of 32 students [72, 75, 80, 68, 78, 71, 73, 74, 76, 69, 70, 77, 79, 67, 81, 68, 72, 75, 70, 69, 71, 74, 73, 77, 78, 80, 72, 69, 70, 75, 76, 71]. Is the sample average significantly different from 70 (the passing score)?
10. Use a Two-Sample Z-test for Population Proportion to compare the rates of on-time delivery for Service A (40 out of 50 deliveries on time) and Service B (30 out of 40 deliveries on time).
11. Use a One-Sample Z-test for Population Proportion to test if the recovery rate for a sample of 22 out of 25 people recovering within 3 days is faster than the national average recovery rate of 0.80.