Significance Tests and Confidence Intervals

Overview Significance Tests

For all Tests and Confidence Intervals applies: first check the conditions that must be met to allow the use of the test.

Conditions for Tests and CI’s concerning a Mean

1. Random Samples or Randomized Experiment
2. 10% condition; this condition is not relevant in case of a randomized experiment
3. normal distribution/ large sample (n>=30) condition

Conditions for Tests and CI’s concerning a Proportion

1. Random Samples or Randomized experiment
2. 10% condition; this condition is not relevant in case of a randomized experiment
3. LC condition; np \(\ge\) 10 and n(1-p) \(\ge\) 10.

In case of two-sample CI’s or tests, check the conditions for both populations/groups separately.

Overview of the STATS —> TESTS menu on the TI84

1. Z-test

The Z-test is used for a 1-sample test about a population mean in case the population standard deviation is known.

H₀: \(\mu\) = \(\mu_0\)
H_A: \(\mu \ne \mu_0\) or \(\mu < \mu_0\) or \(\mu > \mu_0\)

The sample statistic used is a z-value; \(z = \frac{\bar{x}-\mu_0}{\frac{\sigma}{\sqrt{n}}}\)

Report: the name of the test used, z-value, the P-value and the conclusion in the context.

2. T-test

The T-test is used for a 1-sample test about a population mean in case the population standard deviation is unknown.

H₀: \(\mu\) = \(\mu_0\)
H_A: \(\mu \ne \mu_0\) or \(\mu < \mu_0\) or \(\mu > \mu_0\)

The sample statistic used is a t-value with df = n-1; \(t = \frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}\)

Report: the name of the test used, t-value, the P-value and the conclusion in the context.

Note. In a scientific report, the degrees of freedom of the t-distribution used will also be reported; unfortunately the calculator doesn’t provide this output.

3. 2-SampZTest

A 2-Sample Z-Test is used for a significance test about the difference between two population means in case the population standard deviations are known.

H₀: \(\mu_1 - \mu_2 = 0\)
H_A: \(\mu_1 - \mu_2 \ne 0\) or \(\mu_1 - \mu_2 < 0\) or \(\mu_1 - \mu_2 > 0\)

The sample statistic used is: z-value; \(z = \frac{(\bar{x_1} - \bar{x_2}) - 0}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\)

Report: Report: the name of the test used, z-value, the P-value and the conclusion in the context.

4. 2-SampTTest

A 2-Sample Z-Test is used for a significance test about the difference between two population means is case the population standard deviations are known.

H₀: \(\mu_1 - \mu_2 = 0\)
H_A: \(\mu_1 - \mu_2 \ne 0\) or \(\mu_1 - \mu_2 < 0\) or \(\mu_1 - \mu_2 > 0\)

The sample statistic used is: t-value; \(t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)

Report: Report: the name of the test used, t-value, the P-value and the conclusion in the context.

5. 1-PropZTest

The Z-test is used for a 1-sample test about a population proportion.

H₀: \(p\) = \(p_0\)
H_A: \(p \ne p_0\) or \(p < p_0\) or \(p > p_0\)

The sample statistic used is a z-value, \(z = \frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\)

Report: the name of the test used, z-value, the P-value and the conclusion in the context.

6. 2-PropZTest

A 2-Sample Proportion Z-Test is used for a significance test about the difference between two population proportions.

H₀: \(p_1 - p_2 = 0\)
H_A: \(p_1 - p_2 \ne 0\) or \(p_1 - p_2 < 0\) or \(p_1 - p_2 > 0\)

Te sample statistic used is a z-value,

\(z = \frac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}}\)

Report: the name of the test used, z-value, the P-value and the conclusion in the context.

7. ZInterval

A Z-interval is a Confidence Interval for an unknown Population Mean in case the Population Standard Deviation is known.

Formula: \(CI = \bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}\)

Report: Confidence Level, the borders of the CI and interpretation in the context.

8. TInterval

A T-interval is a Confidence Interval for an unknown Population Mean in case the Population Standard Deviation is unknown.

Formula: \(CI = \bar{x} \pm t \times \frac{s}{\sqrt{n}}\) The t-value comes from the t distribution with df = n-1.

Report: Confidence Level, the borders of the CI and interpretation in the context.

9. 2-SampZInt

A two Sample Z Interval is a confidence interval for the difference between two population means in case the standard deviations of the two populations are known.

Formula: \(CI = \bar{x}_1 - \bar{x}_2 \pm z \times \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\)

Report: Confidence Level, the borders of the CI and interpretation in the context.

0. 2-SampTInt

A two Sample Z Interval is a confidence interval for the difference between two population means in case the standard deviations of the two populations are unknown.

Formula: \(CI = \bar{x}_1 - \bar{x}_2 \pm z \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)

Report: Confidence Level, the borders of the CI and interpretation in the context.

A. 1-PropZInterval

A Z-interval is a Confidence Interval for an unknown Population Proportion.

Formula: \(CI = \hat{p} \pm z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{{n}}}\)

Report: Confidence Level, the borders of the CI and interpretation in the context.

B. 2-PropZInterval

A two Proportion Z Interval is a confidence interval for the difference between two population proportions.

Formula: \(CI = \hat{p_1} - \hat{p_2} \pm z \times \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}\)

Report: Confidence Level, the borders of the CI and interpretation in the context.