Statistical Sciences 2035 Study Guide - Quiz Guide: Standard Deviation, Confidence Interval, Statistical Parameter
7 Oct 2018
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Accepting or Rejecting the Null Hypothesis
For testing a null hypothesis, we first calculate the standard error of the mean:
SE(M)=𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
√𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒
Then we take a look at the null hypothesis and the value that is supposed to support it.
The next step is to see how many standard errors the value is away from the mean.
𝑍 = 𝑋 − 𝑀𝑒𝑎𝑛
𝑆𝐸(𝑀)
Any value below 1.96 or above -1.96 has a probability to by chance of 5% = 0.05 or less.
Thus, for a 5 % level of significance the null hypothesis is accepted. Any value above
1.96 leads to the rejection of the null hypothesis in a non-directional test. For
directional tests and 5% level of significance any value above 1.65Z will lead to
rejection, for 1% level of significance it is 2.33Z.
STATISTICAL ERRORS: TYPE I AND TYPE II
There are two types of error that researchers are concerned with: Type I and Type II.
Type I - Rejecting the null when it is true. The probability of rejecting a true null
hypothesis is equal to the alpha level (significance level). (“active error”)
- A Type I error occurs when the results of research show that a significant difference
exists but in reality there is no difference. Therefore, they are determined by the
alpha level and are therefore under the investigator’s control. This is directly related
to alpha in that alpha was likely set too high and therefore lowering the amount of
acceptable error would reduce the chances of a Type I error.
Type II - The probability of accepting the null hypothesis when it is false. (“passive
error”)
- Type II error, or beta ß, the retention of the null hypothesis when it is false. In other
words, the greater the chances of a Type I error, the less likely a Type II error, and
vice versa, because Lowering the amount of acceptable error to 1%, however,
increases the chances of this error.
The selection of a particular level of significance should be made before the data is
analyzed, just as anyone who bets on the horses must do so before the race is run.
Document Summary
For testing a null hypothesis, we first calculate the standard error of the mean: Then we take a look at the null hypothesis and the value that is supposed to support it. The next step is to see how many standard errors the value is away from the mean. Any value below 1. 96 or above -1. 96 has a probability to by chance of 5% = 0. 05 or less. Thus, for a 5 % level of significance the null hypothesis is accepted. 1. 96 leads to the rejection of the null hypothesis in a non-directional test. For directional tests and 5% level of significance any value above 1. 65z will lead to rejection, for 1% level of significance it is 2. 33z. There are two types of error that researchers are concerned with: type i and type ii. exists but in reality there is no difference. Type i - rejecting the null when it is true.
