Statistics 2 summary
Populations are characterized by parameters, such as the mean, the median, the standard deviation
and the proportion. Methods for making inferences about parameters fall into two categories:
estimate the value of the parameter or test a hypothesis.
The point estimate of μ is sample mean y (unbiased estimator). For a large n, y will be normally
distributed with a mean μ and a standard error σ/√n (central limit theorem). An interval estimate of
μ is the probability of y falling in the interval with a level of confidence of 0.95 (for example). .95 is
also called the confidence coefficient.
y−μ 0
The student’s t-test can be used when σ is unknown. t= , with df= n-1, and as n increases, the
s/√ n
distribution of t approaches the distribution of z. The z-test is used for a known σ, with a df=inf. The
tα values can be found in table 2, when looking at the appropriate df. Confidence interval: y ± tα/2 *
s/√n.
A statistical test needs a null hypothesis and a research hypothesis. The test consists of 8 steps:
1. Null hypothesis (H0) and research hypothesis (Ha)
2. Test statistics T.S. (the decision to state whether or not the data support the research
hypothesis)
3. T-distribution under H0 (define df).
4. T-distribution under Ha
5. Type of population mean test
6. Outcome of test statistic t
7. Outcome of p-value
8. Conclusion in (non)statistical terms
A rejection region contains the values of y that support the research hypothesis and reject H0. Type 1
error: rejecting H0 when it is true (based on α). Type 2 error: accept H0 when it is false and Ha is true.
A one-tailed test has the rejection region located in only one tail of the y distribution (for example
when μ>520). For a two-tailed test, Ha: μ ≠ 520, and both tails of the rejection region are included.
Tests for a population mean can be right-tailed (Ha: μ>520), left-tailed (μ<520) or two-tailed (μ ≠
520).
The level of significance (p-value) is defined as the probability of obtaining a value of T.S. that is likely
to reject H0. If p<α, reject H0, and for p>α, fail to reject H0.
Two independent random variables can be normally distributed with μ= μ1-μ2 and σ 2 = σ12 + σ22. The
standard error is σ = √( σ12/n1 + σ12/n2).
For this two-independent t-test, equal variances can be assumed, giving the following:
Sp2 is the weighted average of the sample variances, to determine the unknown σ. For this, a
hypothesis test can also be conducted with a μ= μ1-μ2. The two samples need to be independent
and randomly selected. Dependence can also be the result of spatial correlation.
If there are no equal variances, another test can be performed.
Populations are characterized by parameters, such as the mean, the median, the standard deviation
and the proportion. Methods for making inferences about parameters fall into two categories:
estimate the value of the parameter or test a hypothesis.
The point estimate of μ is sample mean y (unbiased estimator). For a large n, y will be normally
distributed with a mean μ and a standard error σ/√n (central limit theorem). An interval estimate of
μ is the probability of y falling in the interval with a level of confidence of 0.95 (for example). .95 is
also called the confidence coefficient.
y−μ 0
The student’s t-test can be used when σ is unknown. t= , with df= n-1, and as n increases, the
s/√ n
distribution of t approaches the distribution of z. The z-test is used for a known σ, with a df=inf. The
tα values can be found in table 2, when looking at the appropriate df. Confidence interval: y ± tα/2 *
s/√n.
A statistical test needs a null hypothesis and a research hypothesis. The test consists of 8 steps:
1. Null hypothesis (H0) and research hypothesis (Ha)
2. Test statistics T.S. (the decision to state whether or not the data support the research
hypothesis)
3. T-distribution under H0 (define df).
4. T-distribution under Ha
5. Type of population mean test
6. Outcome of test statistic t
7. Outcome of p-value
8. Conclusion in (non)statistical terms
A rejection region contains the values of y that support the research hypothesis and reject H0. Type 1
error: rejecting H0 when it is true (based on α). Type 2 error: accept H0 when it is false and Ha is true.
A one-tailed test has the rejection region located in only one tail of the y distribution (for example
when μ>520). For a two-tailed test, Ha: μ ≠ 520, and both tails of the rejection region are included.
Tests for a population mean can be right-tailed (Ha: μ>520), left-tailed (μ<520) or two-tailed (μ ≠
520).
The level of significance (p-value) is defined as the probability of obtaining a value of T.S. that is likely
to reject H0. If p<α, reject H0, and for p>α, fail to reject H0.
Two independent random variables can be normally distributed with μ= μ1-μ2 and σ 2 = σ12 + σ22. The
standard error is σ = √( σ12/n1 + σ12/n2).
For this two-independent t-test, equal variances can be assumed, giving the following:
Sp2 is the weighted average of the sample variances, to determine the unknown σ. For this, a
hypothesis test can also be conducted with a μ= μ1-μ2. The two samples need to be independent
and randomly selected. Dependence can also be the result of spatial correlation.
If there are no equal variances, another test can be performed.
