Module 6: The single sample t-test

MBB2

The single sample z-test

• we saw in our module that we can calculate a z-score for our sample mean to determine if it is expected or unusual, assuming the null hypothesis is true

• z= M - u/ sigma_m

• where: z = the z-score for our sample mean; M = our sample mean; u= population mean; and sigma_m= standard error of the mean. Note that population standard deviation is known

Population vs Sample Standard Deviation

• However, population standard deviation is rarely known

• In instances when we don’t know the population standard deviation, we can’t use a z-test and the normal distribution to assess our sample mean. That’s okey! Instead, we can use a t-test and the ‘t-distribution’

• We use sample standard deviation as an estimate of population standard deviation. Single sample t-tests and z-tests are very similar otherwise

z-test vs t-test

• z-test standard error ( we know sigma)

  • sigma_m=sigma/sqareroot(n)

• z-test formula

  • z=M-u/sigma_m

• t-test standard error (we estimate sigma with s)

  • s_M=s/sqareroot(n)

• t-test formula

  • t=M-u/s_M

z-test vs t-test

• Almost all aspects of the process are the same when conducting either a single sample t-test or z-test

• We still use Null Hypothesis Significance Testing.

• We still assign a value that indicates no effect to the null hypothesis, and proceed assuming the null is true

• We still apply an alpha level of 5% and determine the probability of our mean occurring. The result determines whether the null hypothesis is rejected or not

One difference about the t-distribution

• In the t-distribution, the critical limit corresponding to our alpha level of 5% will not be fixed at ± 1.96 as it was with the z-test and normal distribution

• Instead, the t-distribution requires that we consider sample size and degrees of freedom (df), when determining the probability of the sample mean.

• Degrees of Freedom are one less than our sample size for a single sample t-test (n-1)

• The central limit varies along with df

Example: Does Head injury Affect IQ?

• Imagine that we have a sample of 10 people with an acquired head injury. Has IQ been affected

• The null hypothesis would be that head injury has not affected IQ. H₀:u=100

• Sample data: M= 94.20, s= 6.16, n= 10

• Let’s calculate a t-score for our sample mean

• t= M-u/s_M

• The probability of our sample mean occurring is low. Reject the null hypothesis

• This evidence suggests that the population of people with this acquired brain injury have lower IQ