T-Tests and Z-Tests

Comparing Two Means

• Utilizes a t-test to compare two groups means where the outcome variable is interval scale or higher, and the predictor is binary.

T-Tests

• There are different types of t-tests depending on the question:
  • One-sample t-tests (Section 13.2)
  • Independent samples t-tests (Sections 13.3 and 13.4)
  • Paired samples t-tests (Section 13.5)
• Cohen’s d measures effect size for a t-test (Section 13.8)
• Assumptions of t-tests and remedies will be focused on in later chapter

One-Sample Z-Test

• A one-sample z-test is rarely used but serves as a stepping stone to the t-test.

Inference Problem Addressed by the Z-Test

• Dr. Zeppo grades intro stats on a curve with an average of 67.5 and a standard deviation of 9.5.
• The grades of 20 students also taking psychology are collected to determine if they score differently.
• The sample mean for the psychology students grades X̄ = 72.3.
• Assess whether the population mean µ for psychology student grades is 67.5.

Constructing the Hypothesis Test

• Null hypothesis (H0): population mean µ for psychology student grades is 67.5.
• Alternative hypothesis (H1): the population mean is not 67.5.
• Psychology grades are normally distributed.
• The true standard deviation of the scores σ is known to be 9.5.
• The difference between the sample mean X̄ and the population mean µ₀ will be the main diagnostic test statistic.
• Compare sample mean X̄ to the population mean, so calculate X̄ - µ0.
• If the quantity is close to 0, it favors the null hypothesis but if it is far away from 0, it is less likely retain
• In this scenario, X ~ Normal(µ₀, σ2).

Sampling Distribution

• The sampling distribution of the mean X̄ is also normal with a mean µ.
• The standard deviation of the sampling distribution, called the standard error of the mean SE(X̄), is σ/√N.
• If the null hypothesis is true: X̄ ~ Normal(µ₀, SE(X̄)).

Standardization

• Convert X̄ into a standard score zX̄ = (X̄ - µ0) / SE(X̄).
• The test statistic is the z-score zX̄.
• Like all z-scores, it has a standard normal distribution: zX̄ ~ Normal(0, 1).
• The z-statistic indicates how many standard errors X̄ is from µ0.
• Rejection regions for the z-test are always the same, as illustrated in Figure 13.3

Z Table

• In the days where calculations were done by hand, someone could publish a table like this:
  • Desired α level, Two-sided test, One-sided test
  • 0.1,. 1.644854, 1.281552
  • 0.05, 1.959964, 1.644854
  • 0.01, 2.575829, 2.326348
  • 0.001, 3.290527, 3.090232
• The desired α level determined the critical Z value

Worked Example with Grades Data

• Calculate sample mean

  • sample.mean <- mean(grades)
  • sample.mean = 72.3

• Create mu.null variable for null hypothesis population mean

  • mu.null <- 67.5

• Create sd.true corresponding to known standard deviation

  • sd.true <- 9.5

• Create N variable for sample size using the function length()

  • N <- length(grades)
  • N = 20

• Now the (true) standard error:

  • sem.true <- sd.true / sqrt(N)
  • sem.true = 2.124265

• Now find the zscore

  • z.score <- (sample.mean - mu.null) / sem.true
  • z.score = 2.259606

• Since 2.26 is bigger then 1.96, but smaller than 2.58 we can say there is a significant effect here, which we would write up like said:

  • "With a mean grade of 73.2 in the sample of psychology students, and assuming a true population standard deviation of 9.5, we can conclude that the psychology students have significantly different statistics scores to the class average (z = 2.26, N = 20, p < .05)."

• Can get p-value using the pnorm() function, which gets the area under the curve below or above specified numbers

  • For upper tail:

    • upper.area <- pnorm( q = z.score, lower.tail = FALSE )
    • upper.area = 0.01192287

  • For lower tail:

    • lower.area <- pnorm( q = -z.score, lower.tail = TRUE )
    • lower.area = 0.01192287

  • The p-value becomes

    • p.value = upper.area + lower.area
    • p.value = 0.02384574

Z-Test Assumptions

• Normality: population distribution is normal.
• Independence: observations in the data set aren't correlated.
• Known Standard Deviation: the true standard deviation of the population is known.
• Because of the unrealistic nature of the 3rd assumption, we can use the t-test

One-Sample T-Test

• Since its not safe to assume psychology students have the same standard deviation as other students, we cannot assume that i know the true value of σ

• Use sd(grades) as a substitute

• However, since we are now relying on an estimate, we must be careful about our new estimate

• To accommodate to fact that we arent completely sure what the true standard deviation is, we use the ttest

• We will refer to the test statistic as a t-statistic

• The t-statistic uses same formula

  • t= (X̄ - μ) / (ô/√N)

Distribution

• The sampling distribution turns into a t-distribution with df = N-1
• A t-distribution is very similar to a normal distribution, but has "heavier" tails
• For large N, the two distributions are identical

The T.Test() Function

• The function is very versatile

• Needs the the 1sr package

• The code to run the test is

  • oneSampleTTest(x=grades, mu=67.5)

Output Explanation

• There are many descriptive statistics
• There are specified hypotheses
• Test results show the t-statistic, df, p-value as well as confidence, and Cohens D

Assumptions

• The T-Test makes the following assumptions:
• Normality - the data is assumed to have a normal distribution
• Independence - The observations in the sample are generated independently of one another