Post on 30-Dec-2015
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Distribution of a sample: The way in which the individual scores within a sample are spread out in relation to the mean.
Sampling Distribution: The way in which sample means, drawn from the same population, spread out from the population mean.
Measure of Variability:
Standard error - the average amount that sample means differ from the population mean.
x
We can use the normal distribution to predict the likelihood of randomly selecting samples with particular means.
= 45 = 10 N = 16
What is the likelihood that a sample, selected at random from this population, would have a mean of 43 or less?
Reminders about Z table:
a column: Z score
b column: prob. between mean and Z
c column: prob. beyond z into the tail
You develop an expensive training program for athletes.
You say that your program reduce event times for runners.
The convention is that we will assume that a sample is different from the population mean for reasons other than chance when the probability of randomly selecting the sample is less than or equal to .05 (5%)
Null hypothesis: Mathematical statement that describes the population in the absence of any treatment.
Ho : = 50
Alternative hypothesis: The mathematical statement that describes the population if the treatment given would be expected to have an effect and therefore change the population.
H1 : 50
If an outcome is statistically significant we reject the null hypothesis and accept the alternative.
Why is the 5% split into two parts?
H1 : 50
This doesn’t suggest the direction of an unusual outcome.
Samples means much less than the population mean of 50 as well as samples means much greater than 50 would be unusual.
This is know as a two tailed test.
Both tails of the distribution are included as possible outcomes in the hypothesis statement.
Assume a teacher knows that students generally have an average score on a test of 85. The standard deviation of all of the students has been 7 points.
Ho : = 85
H1 : 85
The teacher uses a new curriculum this year and finds that the class (N = 20) has an average on the test of 81.
Has this curriculum made a difference in the performance of the class?
From the Z table:
Prob. of a Z score between 0 and
2.56 is .4948 (col. B)
The prob. of a Z score beyond 2.56 is .0052 (col. C)
Conclusion:
Reject the null hypothesis.
This curriculum probably caused the class to perform differently than the population as a whole.
It is possible (although unlikely) that a sample that had a mean of as low as 81 could have been selected at random from the population.
In other words, it is possible (although unlikely) that the curriculum had no effect on the sample.
Type two error (beta error, )
Accept the null hypothesis (saying that the independent variable has no effect) when the null is false.
Z score where 2.5% of the distribution lies in the tail:
Z = + 1.96
Critical value for a two tailed test.
Z score where 5% of the distribution lies in the tail:
Z = + 1.65
Critical value for a one tailed test.