Part A: Hypothesis Testing

1. Null hypothesis:

The null hypothesis states that there is no significant difference between the mean scores of two independent groups.

2. Research hypothesis:

The research hypothesis states that there is a significant difference between the mean scores of two independent groups.

3. Type I error:

Type I error, also known as a false positive, occurs when the null hypothesis is incorrectly rejected. In other words, it indicates that there is a significant difference between the mean scores of two groups when, in reality, there is no difference.

4. Type II error:

Type II error, also known as a false negative, occurs when the null hypothesis is incorrectly accepted. It indicates that there is no significant difference between the mean scores of two groups when, in reality, there is a difference.

5. Statistical significance:

Statistical significance refers to the likelihood that the observed difference between two groups’ mean scores is not due to chance. A statistically significant result indicates that the observed difference is unlikely to have occurred by random variation alone.

6. p-value:

The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. A smaller p-value suggests stronger evidence against the null hypothesis, leading to its rejection in favor of the research hypothesis. Typically, a p-value of less than 0.05 is considered statistically significant.

Part B: One-Sample t-test

In a one-sample t-test, a single group is compared to a known population mean. The goal is to determine if there is a significant difference between the mean of the group and the population mean.

Assumptions:

– The variable of interest is normally distributed in the population.

– The observations are independent.

– The observations within the group are random samples from the same population.

– The population standard deviation is unknown.

Steps for conducting a one-sample t-test:

1. State the null and research hypotheses.

2. Select an appropriate significance level (alpha).

3. Collect data and calculate the sample mean and sample standard deviation.

4. Calculate the test statistic using the formula:

t = (sample mean – population mean) / (sample standard deviation / sqrt(sample size))

5. Determine the critical value or p-value using the t-distribution table or statistical software.

6. Compare the test statistic to the critical value or p-value to make a decision.

7. Interpret the results and draw conclusions.

Part C: Independent Samples t-test

In an independent samples t-test, the means of two independent groups are compared. The goal is to determine if there is a significant difference between the means of the two groups.

Assumptions:

– The variable of interest is normally distributed in each population.

– The observations are independent within and between the groups.

– The variances of the populations are equal (homogeneity of variances assumption).

Steps for conducting an independent samples t-test:

1. State the null and research hypotheses.

2. Select an appropriate significance level (alpha).

3. Collect data and calculate the sample means and sample standard deviations for each group.

4. Calculate the pooled standard deviation using the formula:

sp = sqrt(((n1-1)s1^2 + (n2-1)s2^2) / (n1 + n2 – 2))

where n1 and n2 are the sample sizes of the two groups, and s1 and s2 are the sample standard deviations.

5. Calculate the test statistic using the formula:

t = (sample mean 1 – sample mean 2) / (sp * sqrt(1/n1 + 1/n2))

6. Determine the degrees of freedom using the formula:

df = n1 + n2 – 2

7. Determine the critical value or p-value using the t-distribution table or statistical software.

8. Compare the test statistic to the critical value or p-value to make a decision.

9. Interpret the results and draw conclusions.

In conclusion, hypothesis testing is a powerful tool used in statistical analysis to determine whether there is a significant difference between groups or variables of interest. One-sample t-test and independent samples t-test are commonly used approaches to compare means. By carefully formulating null and research hypotheses, selecting an appropriate significance level, and following the steps outlined in this practice assignment, researchers can effectively analyze their data and draw meaningful conclusions.