Chapter 7: Hypothesis Testing with z Tests – Mini-Textbook Study Notes

Hypothesis Testing with z Tests

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. The z test is a specific hypothesis test used when the population mean and standard deviation are known.

The z Table and Standardization

Benefits of Standardization

• Standardization allows for fair comparisons between different data sets by converting raw scores into a common scale (z scores).

• The z table provides the percentage of scores between the mean and a given z score, facilitating the interpretation of results.

Raw Scores, z Scores, and Percentages

• To determine the percentage associated with a given z statistic:

  1. Convert the raw score to a z score using the formula: $z = \frac{X - \mu}{\sigma}$ where $X$ is the raw score, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

  2. Look up the area in the z table.

• The z table presents the area between the mean and z, and beyond the mean and z.

From Percentages to z Scores

• To find a z score from a given percentage:

  1. Use the z table in reverse, converting a percentage into a z score.

  2. Convert the z score to a raw score using: $X = z\sigma + \mu$

Excerpt from the z Table

Z	% Between Mean and z
0.97	33.40
0.98	33.65
0.99	33.89
1.00	34.13
1.01	34.38
1.02	34.61

Normal Curve and the Standardized z Distribution

Sketching the Normal Curve

• Helps keep concepts clear and minimizes errors.

• Serves as a practical reference and condenses information visually.

The Standardized z Distribution

• The normal distribution is symmetric and bell-shaped.

• Key percentages under the curve: - About 68% of data falls within ±1 z - About 95% within ±2 z - About 99.7% within ±3 z

Calculating Percentiles and Areas

For Positive z Scores

• Percentile for a positive z score: Area to the left of the z value.

• Percentage above a positive z score: Area to the right of the z value.

• Percentage at least as extreme as the z score: Sum of the areas in both tails beyond ±z.

For Negative z Scores

• Percentile for a negative z score: Area to the left of the z value.

• Percentage above a negative z score: Area to the right of the z value.

• Percentage at least as extreme as the z score: Sum of the areas in both tails beyond ±z.

Calculating a Score from a Percentile

• Find the z score corresponding to the given percentile using the z table.

• Convert the z score to a raw score using: $X = z\sigma + \mu$

The z Table and Distributions of Means

z Statistic for a Group

• When analyzing groups, means are used rather than individual scores.

• Samples of many scores are studied, and the mean and standard error for the distribution of means are calculated before the z statistic is determined.

• The standard error of the mean is: $\sigma_M = \frac{\sigma}{\sqrt{N}}$ where $N$ is the sample size.

Assumptions and Steps of Hypothesis Testing

Assumptions

• Assumptions are characteristics about a population necessary for accurate inferences.

The Three Assumptions	Breaking the Assumptions
1. Dependent variable is on a scale measure	Usually OK if the data are not clearly normal or ordinal
2. Participants are randomly selected	OK if we are cautious about generalizing
3. Population distribution is approximately normal	OK if the sample includes at least 30 scores

Parametric Versus Nonparametric Tests

• Parametric tests: Inferential statistical tests based on assumptions about a population (e.g., normality).

• Nonparametric tests: Inferential statistical tests not based on assumptions about the population.

The Six Steps of Hypothesis Testing

1. Identify the populations, distribution, and assumptions, and choose the appropriate hypothesis test.

2. State the null and research hypotheses, in both words and symbolic notation.

3. Determine the characteristics of the comparison distribution.

4. Determine the critical values, or cutoffs, that indicate the points beyond which we will reject the null hypothesis.

5. Calculate the test statistic.

6. Decide whether to reject or fail to reject the null hypothesis.

Statistical Significance

• A finding is statistically significant if the data differ from what would be expected by chance if there were, in fact, no actual difference.

• Statistical significance does not necessarily mean that the finding is important or meaningful.

One-Tailed Versus Two-Tailed Hypothesis Tests

• One-tailed test: The research hypothesis is directional, predicting either an increase or decrease in the dependent variable, but not both.

• Two-tailed test: The research hypothesis does not specify a direction, only that there will be a difference.

Determining Critical Values and Making Decisions

• Critical values are the points on the comparison distribution that define the boundaries for rejecting the null hypothesis.

• After calculating the test statistic, compare it to the critical value to decide whether to reject or fail to reject the null hypothesis.

Research Practices: HARKing and p-Hacking

• HARKing (Hypothesizing After the Results are Known) blurs the distinction between confirmatory and exploratory research.

• p-hacking involves questionable research practices to increase the likelihood of obtaining statistically significant results.