Confidence Intervals and Hypothesis Tests by Zaara Bhatia

Hypothesis Tests

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Purpose: To test a given null hypothesis against an alternative hypothesis.

Alpha Value

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The alpha value (given) is the significance level of the hypothesis test. If the P-Value of the hypothesis test is within the alpha percentage then the result is considered significant enough to reject the null hypothesis.

Alpha is equivalent to 1 - the confidence level

P-Value

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A probability that measures the strength of evidence against the null hypothesis

The smaller the P-value, the greater the evidence is against the null hypothesis. This is because the hypothesis test statistic is further away from the null hypothesis indicating that there is a greater probability Ho is wrong.
Similarly, the greater the P-value, the less evidence there is against Ho.

If P-value > Alpha, then fail to reject Ho

"We do not have enough evidence to reject the null hypothesis"

If P-Value <= Alpha, then reject Ho

"We have strong evidence to conclude that Ho is not true and to reject it"

Distribution of Alpha

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Based on the claim made by the alternative hypothesis, the hypothsis test and alpha value can be distributed in different ways.

If Ha claims that:

P > Po, then the alpha value will be on the right tail

P < Po, then the alpha value will be on the left tail

The above distributions are known as one-tailed

P =/= Po, then the alpha value will be distributed between the right and left tail

This distribution is known as two-tailed

Errors

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Type 1 and 2 errors in Hypothesis Tests cannot be prevented. They happen when the true mean or proportion is unlikely and not predicted by the hypothesis test methods.

In both cases, the errors can have disastrous consequences in real life situations

INFORMATION SECTION

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DECISION CHART

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Question

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Type 1

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This happens when you reject the null hypothesis even though it is true.

The probability of making a Type 1 error is equal to the alpha value since that is the null hypothesis rejection area

Type 2

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This happens when you don't reject the null hypothesis even though it is false

Power is the probability of rejecting a false null hypothesis

Power = 1- beta

Beta - Probability of making a Type 2 Error

Factors
1. Significance Level
As alpha decreases, power decreases therefore beta increases and the probability of Type 2 Error increases

2. Sample Size
As n increases, standard deviation increases, z-score of sample mean/population gets further away from null hypothesis, therefor probability of rejecting null increases (since sample is further away from null)

Confidence Intervals

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A calculated range of values based on sample statistics that we are confident the population parameter is in.

Your confidence level can vary between 90-99%

The equation to calculate a confidence interval is:

CI = statistic +- critical value (standard deviation)

Where the statistic is either the sample mean/proportion or difference in mean/proportion.
The critical value is either the z or t score for the given confidence level
And the standard deviation the equation for either mean standard deviation or population standard deviation.

The example above is a confidence interval for means.

AP Exam Concluding statement:
Depending on the context, you should follow this format when making a CI conclusion.

"We are [confidence level]% confident that that the interval of [confidence interval] contains the true value of the [context]"

Difference between Hypothesis Tests & Confidence Intervals

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MAIN DIFFERENCE:

Hypothesis test compares an alternative hypothesis to a null hypothesis about population parameters.

Confidence intervals use sample statistics to predict a region where the population parameter will be found

Is the null hypothesis a mean or proportion?

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Mean

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Proportion

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Do you know the population standard deviation AND is the sample size more than 30?

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Z-Test

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T-Test

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How many samples do you have?

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One Sample Z-Test for means

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Match Pair Sample Z-Test for means

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Two Sample Z-Test for means

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One Sample T-Test for means

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Match Pair Sample T-Test for means

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Two Sample T-Test for means

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How many samples are there?

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One Sample Z-Test for proportions

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Two Sample Z-Test for proportions

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Comparing Hypothesis Tests & Confidence Intervals

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When comparing confidence intervals and hypothesis tests, we need to make sure that the alpha value and the confidence interval are distributed equally.

One Sample VS Two Sample

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When using two samples in a confidence interval, the only thing that really changes is the statistic and sample error used in the CI equation.

Two Sample Mean

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Note: It says t score here rather than z score because, just like in hypothesis tests, if the standard deviation of the population is unknown OR the sample is not normal, then you use a t-test rather than a z-test for means.

Two Sample Proportion

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Does the question ask you to create a confidence interval?

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Does the question ask you to compare a null and alternative hypothesis?

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Look at the context of the question. Is it about means or proportions?

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Some other type of question

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Mean

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Proportion

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How many samples are there?

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How many samples are there?

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Does the context of your problem require a matched pair design?

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For example: In order to get significant results of the mean student improvement before and after an activity, you need a matched pair design with their results before and after. Where you calculate the difference in performance THEN use those values as your sample to calculate the mean difference in performance.

How many samples do you have?

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Does the context of your problem require a matched pair design?

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One Sample Z-Interval for Proportions

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Two Sample Z-Interval for Proportions

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Does the context of your problem require a matched pair design?

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Two Sample T-Interval for Means

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Do you know the population standard deviation AND is your sample size more than 30?

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Z-Interval

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T-Inteval

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How many samples are there?

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Does the context of your problem require a matched pair design?

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Two Sample Z-Interval for Means

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One Sample T-Interval for Means

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One Sample Z-Interval for Means

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Matched Pair T-Interval for Means

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Matched Pair Z-Interval for Means

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CONFIDENCE INTERVAL

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HYPOTHESIS TEST

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CONNECTIONS

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Confidence intervals and hypothesis tests are both used in tandem to help support our conclusions. For the specific types of problems listed below, you can use both the hypothesis test and confidence interval of corresponding type and color to support your conclusion.

For example, in a mean question with a matched pair design where you need to use the t*score (since the population standard deviation is not provided, or the sample is not normal) then you can use both a match pair t-test and a match pair t-interval to support your conclusion

Matched Pair Sample Z-Score for means

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Two Sample Z-Score for Means

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One Sample Z-Score for Means

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One Sample T-Score for Means

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Matched Pair Sample T-Score for Means

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Two Sample T-Score for Means

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MEAN

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PROPORTION

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Two Sample Z-Score for Proportions

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One Sample Z-Score for Proportions

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THE END

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