Sample Distribution and Proportions

Hypothesis Test

z - Test for a Proportion

Objective

Implement a hypothesis test for one proportion.

z - Test for a Proportion

z - Test for a Proportion

Conditions

Example 1

A pharmaceutical company claims that the vaccine they are developing is effective to 90% of the population. Their final phase of testing involving 1500 randomly selected participants revealed that 1330 showed signs of immunity from the intended disease. Does the pharmaceutical company have strong evidence? Use α=0.05.

Example 1: Solution

Example 1: Solution

Example 1: Solution

Example 1: Solution

Step 4: Decision

Do not reject the null hypothesis.

Example 1: Solution

Step 5: Conclusion

There is enough evidence to support the pharmaceutical company's claim that the vaccine they are developing is effective to 90% of the population.

Example 2

A school psychologist suspects that 20% of high school students spend at least 5 hours per day in online gaming. A randomly selected sample of 200 students found that 56 spend at least 5 hours per day gaming. At 𝛼=0.01, test the claim that more than 20% of high school students spend at least 5 hours per day in online gaming.

Multiple Choice

What are the null and alternative hypotheses?

• $$Ho:\ p\le0.20$$

  $$Ha:\ p>0.20$$

• $$Ho:\ p<0.20$$

  $$Ha:\ p>0.20$$

• $$Ho:\ p=0.20$$

  $$Ha:\ p>0.20$$

• $$Ho:\ p=0.20$$

  $$Ha:\ p<0.20$$

Multiple Choice

What is the direction of the test?

• Left - Tailed

• Right - Tailed

• Two - Tailed

Multiple Choice

What is/ are the critical value/s?

• $$2.576$$

• $$-2.576$$

• $$\pm2.576$$

Open Ended

What is the test statistic value?

Multiple Choice

What is the correct decision?

• Reject the null hypothesis.

• Do not reject the null hypothesis.

Open Ended

Write the conclusion?

Example 3

The quality inspection department requires a certain component to be within 0.01 mm of the specified dimension for the component to pass the quality inspection. A machine used to create the component should produce at least 95% components that pass the quality inspection. If not, the machine would be replaced. A random sample of 1000 components revealed that 937 components passed the quality inspection. Should they replace the machine already? Use 𝛼 = 0.10.

Multiple Choice

What is the null and alternative hypotheses?

• $$Ho:\ p=0.95$$

  $$Ha:\ p<0.95$$

• $$Ho:\ p=0.01$$

  $$Ha:\ p<0.01$$

• $$Ho:\ p=0.95$$

  $$Ha:\ p>0.95$$

• $$Ho:\ p=0.01$$

  $$Ha:\ p>0.01$$

Multiple Choice

What is direction of the test?

• Left - Tailed

• Right - Tailed

• Two - Tailed

Multiple Choice

What is the critical value?

• $$-1.28$$

• $$1.28$$

• $$\pm1.28$$

Multiple Choice

What is the statistic value?

• $$-1.89$$

• $$1.89$$

• $$0.6$$

• $$-0.6$$

Multiple Choice

What is the correct decision?

• Reject the null hypothesis.

• Do not reject the null hypothesis.

Open Ended

Write the conclusion?

Example 3

The largest group of moviegoers by age is the 40- to 59-year-old age group. This group constitutes 32% of the movie-going population. A theater complex randomly surveyed the customers over a three-week period and found that out of 423 surveyed, 170 were 40 to 59 years of age. At the 0.01 level of significance, does this differ from the stated proportion?

Multiple Choice

What is the null and alternative hypotheses?

• $$Ho:\ p=0.32$$

  $$Ha:\ p<0.32$$

• $$Ho:\ p=0.32$$

  $$Ha:\ p<0.32$$

• $$Ho:\ p=0.32$$

  $$Ha:\ p\ne0.32$$

Multiple Choice

What is the direction of the test?

• Left - Tailed

• Right - Tailed

• Two - Tailed

Multiple Choice

What is/ are the critical value/s?

• $$-2.576$$

• $$2.576$$

• $$\pm2.576$$

Multiple Choice

What is the test statistic value?

• $$3.61$$

• $$-3.61$$

• $$3.37$$

• $$-3.37$$

Multiple Choice

What is the correct decision?

• Reject the null hypothesis.

• Do not reject the null hypothesis.

Open Ended

Write your conclusion?

z - Test for a Proportion

1. State the null and alternative hypotheses.

2. Determine the critical values.

3. Compute for the test statistic.

4. Make a decision.

5. Write a conclusion.

References

De Veaux, R. D., Velleman, P. F., & Bock, D. E. (2014). Intro stats (4th Edition). Great Britain: Pearson Education Limited

Navidi, W. & Monk, B. (2019). Elementary statistics, third edition. New York: McGraw-Hill Education