What happens to the standard error of the mean as the sample size increases?

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1. Sampling Distributions

SPSS Assignment

 

The purpose of this SPSS assignment is to examine what happens to the sampling distribution of the mean as the size of each sample for the distribution changes. To do this, we will use SPSS to generate 1000 samples of 25 observations, and then construct four variables (M4, M9, M16, and M25). Each variable represents a sampling distribution and contains sample means based on a different number of observations (e.g., M4 stands for the variable where each mean is based on four observations).  The observations will come from a normal population with µ = 100 and  =

12. The distribution of the resulting set of means can be examined by computing the mean and standard error for each distribution of means.

 

Generating Sampling Distributions using SPSS

 

Generating Sampling Distributions using SPSS

 

1. Open SPSS.

2. Open a new Output window (File > New > Output) and insert your name and student number (Insert > New Title).

3. Go to This PC and follow this path: O:\PsychLab\2101\Lab 4 Sampling\ . Here you will find a syntax file, Sampling.sps, to run your analysis. A syntax file is similar to using the pull-down menus, except here the commands are typed, essentially telling SPSS what to do. Open the syntax file, (i.e., select Sampling.sps) and on the Run menu, select All or manually select all the syntax writting and then click on the Green triangle (Run symbol). You will now have data in your Data Editor and your analysis will appear in your Output window.

4. Export the output to a WORD document. For this assignment there should be three pages of output.

5. The instructions for this are on page 5 of this assignment.

 

Referring to the SPSS problem, answer the following questions.

 

 

We know that the population distribution is normal and that  = 100 and  = 12.

 

1. What should the theoretical mean of each distribution of means be? Why? (2 marks)

 

2. Theoretically, what should the standard error (standard deviation of the means) for each sampling distribution be? (2 marks) Calculate them with the following formula:

 

3. What happens to the standard error of the mean as the sample size increases? (1 mark)

 

4. Describe what happens to the shape of the histograms as we move up in sample size. (1 mark)

 

 

2. Hypothesis Testing about a Single Sample

 

I. SPSS Assignment

Research experiments sometimes involve testing the agility of rats on a rotating cylinder task. The task has an endurance component where a fixed rotation cycle is set, and rats are measured by the length of time they can remain on the cylinder before landing in a cushioned chamber below. One graduate student is told that healthy rats of a certain strain typically remain on the cylinder an average of about fifty seconds. He performs the rotating cylinder task on several Long Evans rats, collecting the data (in seconds) as shown below:

 

38.4 44.6 52.5 48.0 62.8 53.9 50.4 35.0 45.8 50.1 55.1

65.6 55.5 49.5 50.9 44.3 75.8 47.9 76.2 46.2 56.3

 

Is there evidence that the mean endurance time for this task is 50 seconds? Use α = 0.05. (12 marks)

 

Instructions: Using SPSS, determine if there is sufficient evidence to suggest that the mean endurance time for this task is 50 seconds. The work to this question should include the SPSS output and your handwork. You will obtain the SPSS output by following the instructions provided below. For your handwork, follow the hypothesis testing outline steps discussed in the lab. Specifically, your answer should include:

 State the statistical hypotheses (using the appropriate notation).

 Indicate whether the alternative hypothesis is non-directional or directional (and how you know).

 Set the decision rules, which will include:

o State alpha

 

o Compute the degrees of freedom

o Providing the critical value for the test statistic

o Sketch the curve and label the rejection region

o State the decision rules

 The SPSS output

 On the output, circle and label (using the proper notation) the following quantities:

o Sample mean

o Standard error of the mean

o Population mean

o Test statistic

 Calculate the test statistic. You can do this by using the values of the sample mean and standard error of the mean from the SPSS output (you do not have to calculate the sample mean and standard error of the mean from the raw data).

 Draw a conclusion (state whether H0 is rejected or not and why you made this decision).

 Interpret the result

Attachments:

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