Hypothesis Testing: T-tests and Confidence Intervals

• Hypothesis testing involves t-tests and confidence intervals.

Goals

• One-sample t-test, including degrees of freedom and effect size, will be covered
• Confidence Intervals
• Using SPSS

One-Sample Z-Test

• States the hypotheses: H₀: μ = 65 and H₁: μ ≠ 65, N = 35, α = 0.05
• A two-tailed z-test is chosen
• Characterize the sampling distribution by calculating the standard deviation (σx), resulting in 0.592
• Determine critical values (Zcrit) using a z table, then map onto the sampling distribution at and between -1.96 and 1.96
• Compute the test statistic: Zobs = (X - μ) / σx, resulting in 3.38
• Compare Zcrit (1.96) and Zobs (3.38)
• A decision is made about H₀ (reject or fail to reject)
• Express the decision in words describing the findings

Substituting for Unknown Population Standard Deviation (σ)

• It is very rare to actually know the population standard deviation (σ)
• You can estimate σ using the sample standard deviation (s)
• Substitute s into equations to conduct a one-sample t-test

Example Exam Question

• The hypothesis tested whether a new defensive driving class would decrease driving violations by teenagers to below the national average (μ = 3).
• Data was collected from 10 drivers between 16 and 19 who participated in the program
• An appropriate statistical test is conducted to determine if the program decreases violations to a level below the average, using a non-directional test

One-Sample T-Test Steps

• States the hypotheses
• Collects data
• Establishes α (.05)
• Sample size (N-1) determines whether to go to t table
• Calculates the SD of the sample
• Characterizes the sampling distribution
• Calculate estimated standard deviation (s_x) of the samp. dist.
• Use t table to determine critical values (t_crit) or α - .05, df = N - 1, then "map" critical values onto the sampling distribution
• Compute the test statistic:
  • t_obs = (X - μ) / sx​
• Compare t_obs and tcrit
• Make a decision about H₀ (reject or fail to reject H₀)
• The decision is expressed in descriptive words

Z vs T Sampling Distribution

• One-sample z test assumes a known population standard deviation, while a one-sample t test assumes it's unknown
• In the z-test, Central tendency is μx​ = μ; variability uses standard error.
• In the t-test, Central tendency remains μx​ = μ; variability uses estimated standard error. Normal if N > 40
• Z-test is normal if N > 30, the t-test is normal is N > 40
• T-distribution does not follow the "68, 95, 99.7 rule"
• Sample size (N - 1) and alpha are critical when using the t table.

Degrees of Freedom

• The number of independent pieces of information that went into calculating the estimate (e.g., a mean)
• The number of scores that are free to vary (or are free of each other)
• For the tic-tac-toe board, 9 spaces = 8 df
• 5 presentation dates = 4 df
• Knowing the mean only requires knowing N-1 of the scores to deduce the last
• N-1 scores help to find tcrit

T Table Points

• Look down the df column to find the sample size
• Look across the row to obtain the .05 value in the "proportion in two tails column" to find tcrit
• With a large sample size, the z and the t distributions have very similar critical values
• Check out the tcrit value of infinity
• Going up the table from infinity, tcrit becomes more extreme
• The more extreme tcrit values (as df gets smaller) indicate for the shape of the t distribution
• For df not listed: finding df in the table that lands on either side of 35, choose the more conservative df

T vs Z Distributions

• T distributions are bell-shaped and symmetrical
• T distributions have fatter tails than the normal bell curve
• Shape depends on df
• As N (sample size) increases, t is more like z
• Thicker tails on the T distrubution (squished down on the top and squirting/pushing more of the scores away from the center and into the tails) = more AUC in is in the tails.

T Distributions

• The non-directional hypothesis has a = .05
• Larger sample sizes allow for one to find '.0500' closer to -1.96 in each tail as opposed to -2.45
• The t to z distribution means that greater proportion of the sampling distribution in tails.

T Distributions and Z Distribution Compared

• t distributions are more variable than z distributions when N is small
• z has σ in the formula for standard error, which is constant
• t has s in formula for standard error--it varies from sample to sample
• Sigma is constant to z distributions so samples with same mean or n will produce the same Zobs
• Estimated standard error varies from sample to sample so a consistent mean for n will produce different Tobs

In Z Distribution

• z formula has sigma in the denominator, which is constant
• Standard error (SEM) doesn't vary from sample to sample as it comes from the population variance.
• Samples with the same mean and N share the a matching z-score.

In T distribution

• T formula has s in the denominator, with S based on the sample variance
• Samples can have the same mean, yet different values of tobs because the estimated standard deviation and therefore the estimated standard error will vary from one sample to another.
• A t distribution will have more variability than the normal z distribution, leading to fatter tails, especially when sample size is SMALL.

T Test Assumptions

• σ is not known
• Sample is independently and randomly selected from population
• Population has a normal distribution--test is robust for violations when N ≥ 10

Effect Size: Cohen's d

• Only calculate d for significant effects!
• How big the difference is between the sample mean and the population mean in standard deviation units
• Differences between z compared to t test is whether pop. or sample is used accordingly
• For Z-Test Cohen's d = (X - μ)/ σ
• For T-Test Cohen's d = (X - μ)/ s
• Effect sizes:
• Small |d| ≤ 0.2
• Medium 0.2 > |d| > 0.8
• Large |d| ≥ 0.8

Measures of Effect Size

• Eta-squared (η²) will be covered with the independent groups t test
• Omega-squared (w²) will not be covered

Estimation

• A sample statistic is used to estimate a population parameter without a null or alternative hypothesis
• Point estimation (the sample mean) gives and unbiased estimate
• Interval estimations (confidence intervals) are a range (around a sample mean), that estimates where the population mean ('comes from') is likely to fall
• Guessing a team’s weight illustrates a confidence interval

Confidence intervals (CIs)

• Range of values expressing confidence in a the true population mean--estimate μ with X, use standard error for observed z or t value
• A 95% confidence interval is where α = .05; for 99% CIs, α = .01
• If Cl does not contain the μ specified in Ho, it is unlikely that your sample mean came from the null distribution
• When σχ is known: CI = X ± |Zcrit|(σ)
• When σ is NOT known: CI = X ± |tcrit|(Sx)

Driving Violations Example

• σ is NOT known: CI = X ± |tcrit|(Sx)
• 95% CIs = [1.46, 2.94]
• I'm 95% confident that this sample came from a population where the mean # of driving violations was between 1.46 and 2.94
• Population mean for 𝐻0 (μ = 3) does NOT fall in this range, so Ho is rejected
• Notation: express data in terms of standard +/- or with square brackets surrounding data (+/- terms)
• To be 95% confident- estimate tcrit along with std. errors on respective sides of sample mean

One-Sample T test

• One-tailed: 𝑡 critical value for 𝛼 = .05 with 95% CI: 2.2 ± |2.262|(0.327) = [1.46, 2.94]
• To be 95% confident
• "Critical regions for the one-sample t test are 2.262 standard errors away from the POPULATION mean
• The region is shifted to be around the SAMPLE mean rather than the population mean

Conducting a One-sample T Test in SPSS

• Enter data into a column, naming and labeling the variable
• Go to Analyze/Compare means/One sample t test and enter the variable as test
• Enter the population mean as the "test value" and click "OK"
• Various SPSS resources include: Yellow Book (SPSS instruction packet), Privitera 9.6 SPSS in Focus: One-sample t test (pp. 286-288), Privitera Appendix B-6: 9.1 One-sample t test (p. B-6), and office hours

SPSS Output Points

• Use p. 21 of your SPSS yellow packet for an annotated SPSS output example
• If p < .05, then Ho is rejected

Cohen’s d in SPSS

• The output for Cohen's d is missing from the sample annotation on p. 21 of your yellow book, but please be sure to include it when you do your homework and annotations

Confidence Intervals in SPSS

• SPSS draws an Cl around the difference between X and μ, not around 𝑋
• convert it, just add μ to the lower and upper limits using 3 to calculate 2.2 ±[1.46, 2.94]

One Sample APA Results Points

• Variables are named (IV, DV), along with μ, M,
• The words 'significant' and 'direction' of effect are included
• M and SD are reported for DV
• A 't statement' is listed
• Report the CI around the sample mean

Understanding hypothesis testing using t-tests and confidence intervals. Includes calculating standard deviation, determining critical values, and making decisions about hypotheses. Covers substituting for unknown population standard deviation.