Up Learn – A Level maths (edexcel) – Testing a Population Mean
Hypothesis Testing: Normal Distribution
Here’s a summary of everything you need to know about hypothesis testing for normal distributions at A Level.
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More videos on Testing a Population Mean:
Introduction to Testing a Population Mean (free trial)
Sample Means and Population Means (free trial)
How Sample Size Affects Sample Means (free trial)
Probabilities of Sample Means (free trial)
Modelling the Sample Mean (free trial)
The Standard Deviation of the Sample Mean’s Distribution (free trial)
When Should We Question Our Population Mean? (free trial)
Finding Critical Regions and Critical Values (free trial)
What Are the Hypotheses? (free trial)
Performing a One-Tailed Test for a Population Mean (free trial)
Performing a Two-Tailed Test for a Population Mean (free trial)
Testing a Population Mean with the Standard Normal Distribution (free trial)
Hypothesis Tests
2. Statistics vs Parameters (free trial)
3. The World is Unknowable (free trial)
4. Don’t Treat Parameter Estimates as Final (free trial)
5. Population Proportions (free trial)
6. When Should We Question Our Parameter Estimates? (free trial)
7. Critical Regions (free trial)
8. Critical Regions With Two Tails (free trial)
9. One or Two Tails? (free trial)
10. Critical Values (free trial)
11. Acceptance Region (free trial)
12. Type 2 Errors (free trial)
13. Type 1 Errors (free trial)
14. Remembering Which is Which (free trial)
15. Significance (free trial)
16. Significance Levels (free trial)
17. Significance Levels with Two Tails (free trial)
18. Actual Significance Levels (free trial)
19. Common Significance Levels (free trial)
20. Two Ways of Setting a Significance Level (free trial)
21. Finding Critical Regions (free trial)
22. The Term ‘Hypothesis Testing’ (free trial)
23. Population Parameters and Test Statistics (free trial)
24. What are Hypotheses? (free trial)
25. Null and Alternative Hypotheses (free trial)
26. Statistical Hypothesis Testing (free trial)
27. One-Tailed Tests and Two-Tailed Tests (free trial)
28. Representing the Null and Alternative Hypotheses (free trial)
29. Performing a One-Tailed Test (free trial)
30. A Second Way of Performing a One-Tailed Test (free trial)
31. Performing a Two-Tailed Test (free trial)
32. The Miraculous Dead Salmon (free trial)
2. Sample Means and Population Means (free trial)
3. How Sample Size Affects Sample Means (free trial)
4. Probabilities of Sample Means (free trial)
5. Modelling the Sample Mean (free trial)
6. The Standard Deviation of the Sample Mean’s Distribution (free trial)
7. When Should We Question Our Population Mean? (free trial)
8. Finding Critical Regions and Critical Values (free trial)
9. What Are the Hypotheses? (free trial)
10. Performing a One-Tailed Test for a Population Mean (free trial)
11. Performing a Two-Tailed Test for a Population Mean (free trial)
12. Testing a Population Mean with the Standard Normal Distribution (free trial)
13. Coding the Sample Means (free trial)
14. Hypothesis Testing With a Coded Sample Mean (free trial)
2. Correlation Coefficients from Samples and Populations (free trial)
3. Probabilities of Sample Correlation Coefficients (free trial)
4. Sample Size Matters (free trial)
5. Testing for Zero Correlation (free trial)
6. The Null and Alternative Hypotheses in PMCC Testing (free trial)
7. The Percentage Points Table (free trial)
8. Performing a One-Tailed Test for a Population PMCC (free trial)
9. Performing a Two-Tailed Test for a Population PMCC (free trial)
10. Where Were All the Distributions? (free trial)
11. Are Older People All Liars? (free trial)
Here’s a reminder of the key points you should know about testing a population mean.
Population means and sample means are often different.
But the bigger the sample, the closer the sample mean is likely to be to the population mean.
If a population is modelled using a normal distribution…
It’s possible to model the probabilities of observing different sample means like this.
Where ‘X bar’ represents the sample mean and n is the sample size.
Then, the variance of the new distribution is this and the standard deviation is this.
If we have an estimate for a population mean, we can test it out using hypothesis testing.
The logic goes: if the estimate is correct, we’re very unlikely to observe some sample means.
So if we take a sample and do observe those unlikely means, we should reject the estimate we currently have.
To find critical regions on a normal distribution, use the inverse normal distribution function on your calculator.
The hypotheses look pretty much the same as when we test using a binomial distribution.
It’s just that, since we’re forming hypotheses about mean values now, we use.
Finally then, the exam may tell you a sample was taken, tell you the significance level and ask you to perform the rest of the hypothesis test.
Here’s what that looks like for a one-tailed test.
First, write down the null and alternative hypotheses.
Second, write down the distribution for the sample mean.
Third, find and state the critical region.
Fourth, compare the test statistic to the critical region.
And fifth, write a conclusion.
An alternative method is to find the probability of observing the sample mean or anything greater,
Compare that to the significance level, and then write the conclusion accordingly.
The working looks very similar for a two-tailed test.
First, write down the null and alternative hypotheses.
Second, write down the distribution for the sample mean.
Third, find and state the critical region, now with two tails.
Fourth, compare the test statistic to the critical region.
And fifth, write a conclusion.
In place of this step, you could find the probability of observing the sample mean or anything greater,
Compare that to half of the significance level, since this is a two-tailed test,
And then write the conclusion accordingly.
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