Session 17 Notes, Section 8-4

Hypothesis Testing:  t Tests (Student t Test)  for Means - Small Samples

In the previous sessions, we learned the classical and P-value methods for test for a mean in the case of "large" samples.  In this session we will explore the testing of the mean when we have "small samples".  We will use both the classical and P-value methods.

At the end of this session you will be able to:

• Test means for small samples using the t test.
  • Example
  • Example

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Test Means for Small Samples Using the t Test (Student t Test).

This test goes by two different names: t Test or Student t Test.

• The t test is a statistical test of the mean of a population and is used when the population is normally or approximately normally distributed, σ is unknown, and the sample size ≤  30.
• The formula for the t test is:
  • 
• The degrees of freedom are d.f. = n–1.\

In the Classical or  Traditional Method  of solving problems with the t test is done using the same method as with the z text EXCEPT that t values are found on the T table (Table F of your textbook) instead of using Z values found on the Z table (Table E of your textbook).

P-values for the t test can only be found for intervals using Table F in your book.  You use the value that is lower than and above the actual value you are looking up to be the interval in which the p-value would lie.

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Example.

• Find the critical value for α= 0.05 with d.f. = 12 for a right-tailed t test.
  • Go to table F page 731
  • Find the 0.05 column in the top row and 12 in the left-hand column.
  • Where the row and column meet, the appropriate critical value is found: it is +1.782
• Find the critical values for α= 0.50 with d.f. = 18 for a two-tailed t test.
  • Go to table F page 731
  • Find the 0.50 column in the row labeled "Two tails", and find 18 in the column labeled "d.f.".
  • The critical values are: +0.688 and -0.688.

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Example.

A job placement director claims that the average starting salary for nurses is $26000. A sample of 14 nurses has a mean of $25560 and a standard deviation of $500. Is there enough evidence to reject the director's claim at α= 0.05?

• Review the steps for hypotheses testing.  Follow them step by step.
• Step 1: State the hypotheses and identify the claim.
  • H₀:  µ = 26000 (claim)  and  H₁: µ not = 26000
• Step 2: Find the critical value.
  • The critical values are +2.16 and -2.16 for = 0.05 and d.f. = 13
• Step 3: Compute the test value.
  • or = -3.29
• Step 4: Make the decision.
  • Reject the null hypothesis, since -3.29 < -2.16. 
• Step 5: Summarize the results.
  • There is enough evidence to reject the claim that the starting salary of nurses is $26000.

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State the five steps used in hypothesis testing (Classical or Traditional Method).

• Step 1 State the hypothesis, and identify the claim.
• Step 2 Find the critical value from the appropriate table.
• Step 3 Compute the test value using t  = (X - μ) /( s/√n).
• Step 4 Make the decision to reject or not reject the null hypothesis.
• Step 5 Summarize the results.

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