The z Table and Hypothesis Testing with z Tests

The z Table

  • Overview of the z Table

    • In Chapter 6, the empirical rule was introduced:

    1. About 68% of scores fall within one z-score of the mean.

    2. About 96% of scores fall within two z-scores of the mean.

    3. Nearly all scores fall within three z-scores of the mean.

    • These are useful guidelines, but for precise calculations, the z-table is used.

    • The full z table is located in Appendix B.1, with an excerpt provided in Table 7-1.

Understanding the Z Table

  • The z table gives percentages of scores between the mean and a given z value.

  • Negative z-statistics can be calculated by changing the sign of their positive counterparts.

  • The normal curve is symmetric, implying that the percentages associated with negative and positive z scores are identical.

  • Example Table Excerpt (Table 7-1)

Raw Scores, z Scores, and Percentages

  • Relationship Between Scores

    • Much like various names can refer to the same individual (e.g., "Christy," "Tina" for "Christina"), z scores, raw scores, and percentile rankings refer to the same statistical concept.

    • The z table facilitates the transition between these types of scores.

    • It serves as a mechanism for stating and testing hypotheses by standardizing different observations onto a common scale.

  • Calculating Percentages with the z Table

    • Step 1: Convert the raw score into a z score.

    • Step 2: Use the z table to find the percentage of scores between the mean and the calculated z score.

    • Note: The z scores in the table are typically positive; negative z scores can be deduced using the symmetry of the normal distribution.

Visual Representation of the Standardized z Distribution

  • The z table allows for the calculation of percentages above and below a specific z score.

  • Symmetry of the normal curve informs us that negative z scores mirror their positive counterparts, providing identical percentages.

Example Calculations

Example 7.1

  • A research team investigated whether shorter children experience poorer psychological adjustment compared to taller peers, potentially justifying treatment with growth hormone (Sandberg et al., 2004).

Methodology

  • The study categorized boys and girls aged 15 years based on height into three classifications:

    1. Short (bottom 5%)

    2. Average (middle 90%)

    3. Tall (top 5%)

  • Data acquired from the Centers for Disease Control provided average heights for 15-year-old boys and girls.

    • Boys:

      • Mean height: 67.00 inches (170.18 cm)

      • Standard deviation: 3.19 inches (8.10 cm)

    • Girls:

      • Mean height: 63.80 inches (162.05 cm)

      • Standard deviation: 2.66 inches (6.76 cm)

Case of Jessica

  • Step 1: Convert Jessica's height (66.41 inches) to a z score using girls' data:

    z=X−μσ=66.41−63.802.66=0.98z = \frac{X - \mu}{\sigma} = \frac{66.41 - 63.80}{2.66} = 0.98

  • Step 2: Search for z = 0.98 in the z table, yielding a percentage of 33.65% between the mean and Jessica's z score.

Derived Percentages

  1. Jessica's Percentile Rank:

    • Total percentage below Jessica: 50% (scores below the mean) + 33.65% = 83.65%

2.) Percentage of Scores Above Jessica:

  • Above scores: 50% (total above mean) - 33.65% = 16.35%

3.) Scores as Extreme as Jessica's:

  • Total extreme scores in both directions: 16.35% + 16.35% = 32.70%

  • Classification:

    • Since 16.35% of 15-year-old girls are taller than Jessica, she is not in the top 5%, so we can classify her as being average in height.

Example 7.2

  • Case of Manuel (Height: 61.20 inches)

Steps to Analyze Manuel's Height

  • Step 1: Convert his height to a z score using the boys' data:

    z=X−μσ=61.20−67.003.19=−1.82z = \frac{X - \mu}{\sigma} = \frac{61.20 - 67.00}{3.19} = -1.82

  • Step 2: Identify z = 1.82 in the z table, yielding 46.56% percentage between the mean and z.

Derived Percentages for Manuel

  1. Manuel's Percentile:

    • Score below: 50% - 46.56% = 3.44%

2.) Percentage of Scores Above Manuel score :

  • Above scores: 50% + 46.56% = 96.56%

3.) Extreme Percentages:

  • Total extreme heights: 3.44% (below) + 3.44% (above) = 6.88%

  • Classification:

    • Manuel's percentile rank of 3.44% places him in the lowest 5% of heights, categorizing him as short.

Example 7.3

  • Exploration of the Relationship between Raw Scores, z Scores, and Percentiles

  • A symbolic representation of autism using a puzzle piece has elicited criticism from individuals with autism who prefer a rainbow infinity symbol (Jessop, 2019).

    • A study aimed to assess negative perceptions linked to the puzzle piece among the general public (Gernsbacher et al., 2018).

    • An Implicit Association Test (IAT) revealed a normal distribution of implicit association scores towards puzzle pieces, where the mean was -0.14, and the standard deviation was 0.51, indicating a negative bias across the sample.

Calculating a Participant's IAT Score at the 63rd Percentile

  1. Determine Raw Score:

    • The score measures above the mean since 63% is greater than 50%.

    • Calculate percentage: 63% - 50% = 13%.

    • Locate 12.93% in the z table, yielding a z score = 0.33.

  1. Convert z Score to Raw Score:

    • Using the formula:

    X=μ+z(σ)X = \mu + z(\sigma)
    Substituting in numbers:

    X=−0.14+(0.33)(0.51)=0.0283X = -0.14 + (0.33)(0.51) = 0.0283

Result Validation

  • The calculated score corresponds correctly as above the mean and aligns with the percentile rank being above 50%.

Conclusion

  • The document demonstrates the utility of the z table for hypothesis testing through systematic steps, examples involving real-world applications, and explicit calculations.

  • Future studies could expand on the application of z-scores to other psychological metrics, enhancing the understanding of height influences and perceptions across different demographics.