Question

14% of all Americans suffer from sleep apnea. A researcher suspects that a lower percentage of those who live in the inner city have sleep apnea. Of the 307 people from the inner city surveyed, 34 of them suffered from sleep apnea. What can be concluded at the level of significance of αα = 0.01?

1. For this study, we should use Select an answer z-test for a population proportion t-test for a population mean
2. The null and alternative hypotheses would be:
   Ho: ? p μ  ? = > < ≠   (please enter a decimal)   
   H₁: ? p μ  ? = < ≠ >   (Please enter a decimal)

3. The test statistic ? t z  =  (please show your answer to 3 decimal places.)
4. The p-value =  (Please show your answer to 4 decimal places.)
5. The p-value is ? ≤ >  αα
6. Based on this, we should Select an answer fail to reject reject accept  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the population proportion is not significantly smaller than 14% at αα = 0.01, so there is sufficient evidence to conclude that the population proportion of inner city residents who have sleep apnea is equal to 14%.
   • The data suggest the populaton proportion is significantly smaller than 14% at αα = 0.01, so there is sufficient evidence to conclude that the population proportion of inner city residents who have sleep apnea is smaller than 14%
   • The data suggest the population proportion is not significantly smaller than 14% at αα = 0.01, so there is not sufficient evidence to conclude that the population proportion of inner city residents who have sleep apnea is smaller than 14%.
8. Interpret the p-value in the context of the study.
   • If the population proportion of inner city residents who have sleep apnea is 14% and if another 307 inner city residents are surveyed then there would be a 6.98% chance fewer than 11% of the 307 residents surveyed have sleep apnea.
   • There is a 14% chance of a Type I error
   • If the sample proportion of inner city residents who have sleep apnea is 11% and if another 307 inner city residents are surveyed then there would be a 6.98% chance of concluding that fewer than 14% of inner city residents have sleep apnea.
   • There is a 6.98% chance that fewer than 14% of all inner city residents have sleep apnea.
9. Interpret the level of significance in the context of the study.
   • There is a 1% chance that the proportion of all inner city residents who have sleep apnea is smaller than 14%.
   • If the population proportion of inner city residents who have sleep apnea is smaller than 14% and if another 307 inner city residents are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all inner city residents who have sleep apnea is equal to 14%.
   • There is a 1% chance that aliens have secretly taken over the earth and have cleverly disguised themselves as the presidents of each of the countries on earth.
   • If the population proportion of inner city residents who have sleep apnea is 14% and if another 307 inner city residents are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all inner city residents who have sleep apnea is smaller than 14%.

Answer 1

(a) z-test for a population proportion.
(b) H₀: p = 0.14, H₁: p < 0.14.



(g) The data suggest the population proportion is significantly smaller than 14% at αα = 0.01, so there is sufficient evidence to conclude that the population proportion of inner city residents who have sleep apnea is smaller than 14%

(h) If the population proportion of inner city residents who have sleep apnea is 14% and if another 307 inner city residents are surveyed then there would be a 6.98% chance fewer than 11% of the 307 residents surveyed have sleep apnea.

(i) If the population proportion of inner city residents who have sleep apnea is 14% and if another 307 inner city residents are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of all inner city residents who have sleep apnea is smaller than 14%.