MCAD Hypothesis Test for a Difference in Two Population Means Worksheet

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Learn by Doing

In this activity you will learn to use StatCrunch to perform a two-sample t-test.

Some features of this activity may not work well on a cell phone or
tablet. We highly recommend that you complete this activity on a
computer.

Here are the directions and grading rubric for the discussion board exercises.

A list of StatCrunch directions is provided at the bottom of this page.

Context

Do undergraduates sleep less than graduate students?

A student conducted a study of sleep habits at a large state
university. His hypothesis is that undergraduates will party more and
sleep less than graduate students. He surveyed random samples of 75
undergraduate students and 50 graduate students. Subjects reported the
hours they sleep in a typical night.

For this hypothesis test, he defined the population means as follows:

  • LaTeX: mu_1


    μ
    1

    is the mean number of hours undergraduate students sleep in a typical night.

  • LaTeX: mu_2


    μ
    2

    is the mean number of hours graduate students sleep in a typical night.

Variables

Hours: typical number of hours a student sleeps each night
Program: undergraduate or graduate

Program is the explanatory variable, and the data is categorical. Hours is the response variable, and the data is quantitative.

Data

Two sample T hypothesis test:

μ1 : Mean of undergraduate
μ2 : Mean of graduate
μ1 – μ2 : Difference between two means
H0 : μ1 – μ2 = 0
HA : μ1 – μ2 < 0
(without pooled variances)

Hypothesis test results:

Difference Sample Diff. Std. Err. DF T-Stat P-value
μ1 – μ2 -0.23333333 0.18963708 106.32776 -1.2304204 0.1106

Prompt

  1. State the null and alternative hypotheses. Include a clear description of the populations and the variable.
  2. Explain why we can safely use the two-sample T-test in this case.
  3. Use StatCrunch to carry out the test.
    Copy and paste the content of in the StatCrunch output window (text and the table) in your initial post.
  4. State a conclusion in the context of this problem.

ANSWER(S): (hint)

  1. Since we want to check whether the data supports the claim that
    undergraduate students sleep less, on average, than graduate students,
    we are testing:
    H0


    H
    0

    : mu sub 1 - mu sub 2 equals 0


    μ
    1



    μ
    2

    =
    0

    H sub a


    H
    a

    : mu sub 1 - mu sub 2 is less than 0


    μ
    1



    μ
    2

    <
    0

  2. We can safely use the two-sample T-test in this case since:

    • Both samples are random, and therefore independent.
    • The sample sizes (75 and 50) are quite large, and therefore we can
      proceed regardless of whether the populations are normal or not.

  3. T = -1.23; P-value = 0.111

    Here is the StatCrunch output depicting these values.

    Two sample T hypothesis test:
    LaTeX: mu_1


    μ
    1

    : Mean of undergraduate
    LaTeX: mu_2


    μ
    2

    : Mean of graduate
    LaTeX: H_0


    H
    0

    : LaTeX: mu_1-mu_2=0


    μ
    1



    μ
    2

    =
    0


    LaTeX: H_a


    H
    a

    : LaTeX: mu_1-mu_2<0


    μ
    1



    μ
    2

    <
    0


    (without pooled variances)

    Hypothesis test results:
    Difference Sample Diff. Sd. Err. DF T-Stat P-Value
    μ1 – μ2 -0.23333333 0.18963708 106.32776 -1.2304204 0.1106

  4. The P-value (0.111) is greater than 0.05. This indicates that
    the observed data is not unusual if undergraduate and graduate students,
    on average, have the same mean for sleep hours. Therefore, the data do
    not provide evidence to reject H0


    H
    0

    . So we cannot conclude that undergraduate students sleep less, on average, than graduate students.

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