Week 1: The Single-Sample z Test


Today we’ll be discussing one of the most commonly used statistical tests and one of the first ones taught in introductory stats: the single-sample z test!

When Would You Use It?
The single-sample z test is a parametric test used in a single sample situation to determine if the sample originates from a population with a specific mean µ. This test is used when the population standard deviation, σ, is known.

What Type of Data?
The single-sample z test requires interval or ratio data.

Test Assumptions

  • The sample is a simple random sample from the population of interest.
  • The distribution underlying the data is normal.

Test Process
Step 1: Formulate the null and alternative hypotheses. The null hypothesis claims that the mean in the population is equal to a specific value; the alternative hypothesis claims otherwise (the population mean is greater than, less than, or not equal to the value specified in the null hypothesis.

Step 2: Compute the z-score. The z-score is computed as follows:

Test 1

Step 3: Obtain the p-value associated with the calculated z-score. The p-value indicates the probability of observing a sample mean as extreme or more extreme than the observed sample mean, under the assumption that the null hypothesis is true.

Step 4: Determine the conclusion. If the p-value is larger than the prespecified α-level, fail to reject the null hypothesis (that is, retain the claim that the mean in the population is equal to the value specified in the null hypothesis). If the p-value is smaller than the prespecified α-level, reject the null hypothesis in favor of the alternative.

Example
The data for this example are n = 400 IQ scores from random sample of individuals ages 18 to 87. The test used to obtain the scores was constructed similar to the Stanford-Binet IQ test, meaning that we can assume that the average IQ score in the population should be 100 and the population standard deviation σ is known and is equal to 15. Let’s test the claim that the population mean is actually different than 100. Set α = 0.05.

H0: µ = 100
Ha: µ ≠ 100

The sample mean is calculated to be 101.864

Computations:

Test 1b

Since our p-value is smaller than our alpha-level (0.012 > 0.05), we reject H0 and claim that the population mean is different from 100.

Example in R

dat=read.table('clipboard',header=T) #'dat' is the name of the imported raw data
mu = 100
sigma = 15
n = 400
xbar = mean(dat)
z = (xbar-mu)/(sigma/sqrt(n))  #z-score                  
pval = (1-pnorm(z))*2          #p-value
                               #pnorm calculates the left-hand area
                               #multiply by two because it is a two-sided test
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