Tests of SignificanceThe “Student’s t test”:
Single Sample t test:- The single sample t test is used to compare a single sample mean with a population mean when is unknown & the sample size is small (< 30).
- The formula for the t statistic is similar in structure to the Z, except that the t statistic uses sample standard deviation.
The One-Sample t TestOne Sample t test: Step by step:
1- State the null hypothesis.
2- Calculate the t value.
3- Calculate the degrees of freedom df = n-1.
4- State the critical value(s) (From ‘the t distribution table’).
5- Draw a conclusion:
Compare the calculated t to the critical values from the t table to determine significance
If the calculated t value is smaller than the critical values
P > 0.05
Accept the null hypothesis
If the calculated t value is equal to or larger than the critical values.
P< 0.05 or < 0.01
o reject the null hypothesis.
The t Test for a Single Sample: ExampleThe mean weight of 9 undernourished 6-year olds children was 17.3 kg with a standard deviation =2.51 kg. If the weight of 6-year olds in the general population is normally distributed with µ=20.9 kg. Determine if the weight of this sample is significantly different from the population of 6-year olds.
Solution: ِِِِ1. State the null hypothesis
The null hypothesis:
There is no difference between the mean weight of the malnourished 6-year olds and the mean weight of 6-year olds in the general population and the observed difference is due to chance or sampling error.
2. Calculate the t value:ِِِِِِِِِِِِ
3. Calculate the df :df = 9-1= 8
4. State the critical values:
From the t distribution table:
The critical values at 8 degrees of freedom are:
df 0.05 0.01
8 2.31 3.36
- Since the calculated t value is larger than the critical tabulated value at 95% level therefore p<0.05.
- The calculated t value is also larger than the critical tabulated value at 99% level therefore p<0.01.
We reject the null hypothesis which means that the mean weight of malnourished 6-year olds children is significantly different from the mean weight of 6-year olds children in the general population.
Two Samples test for independent samples :The test statistic z requires:
the two populations are normal with known variances
both sample sizes are greater than 30, therefore sample variances may be used as estimates to provide an approximate z statistics
ThereforeWhen population variances are unknown& the sample sizes are small( one or both sample sizes are less than 30) the t test can be used :
If it is assumed that the population variances are equal & the samples are taken from two normally or approximately normally distributed populations.
t test for independent samples:The hypotheses are:
H0: µ1 - µ2 = 0
H1: µ1 - µ2 # 0
The test statistic is:
t Test Statistic
Pooled Estimate of the Variancedf = n 1 + n 2 -2
Pooled Estimate of the Variance:- The pooled estimate of the population variance is the average of both sample variances, once adjusted for their degrees of freedom.
1- You know you have made a mistake in calculating the pooled estimate of the variance if it does not come out between the two estimates.
2- You have also made a mistake if it does not come out closer to the estimate from the larger sample.
Summary:- The comparison between two means can be made with the Z test if the samples are independent and the variances are known, or if the variances are unknown but both sample sizes are greater than or equal to 30.
- If the variances are not known or one or both sample sizes are less than 30, then the t test must be used.
t test for dependent samples)Paired samples-t test)Dependent Samples are samples that are paired or matched in some way. Independent Samples are samples that are not related.
Examples of Dependent Samples:
1- Samples in which the same subjects are used in a pre-post situation.
2- Another type of dependent samples is matched samples.
1.Calculate the difference for each of the pairs of data, d.
2. Find the mean of the differences. d.
3. Find the standard deviation of the differences, Sd.
4. Find the estimated standard error of the differences.
5. Find the test value, t.
Test statisticThe hypotheses for this problem are:
H0 : µd = 0
H1 : µd 0
Assuming that the differences between pairs of values are approximately normally distributed.