Question

The average number of cavities that thirty-year-old Americans have had in their lifetimes is 5. Do twenty-year-olds have more cavities? The data show the results of a survey of 13 twenty-year-olds who were asked how many cavities they have had. Assume that the distribution of the population is normal.

4, 7, 4, 6, 5, 4, 5, 5, 5, 5, 5, 4, 4

What can be concluded at the αα = 0.01 level of significance?

1. For this study, we should use Select an answer t-test for a population mean z-test for a population proportion
2. The null and alternative hypotheses would be:

H0:H0:  ? μ p  ? = ≠ < >       

H1:H1:  ? p μ  ? = ≠ < >    

3. The test statistic ? z t  =  (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 that the population mean number of cavities for twenty-year-olds is not significantly more than 5 at αα = 0.01, so there is insufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is more than 5.
   • The data suggest the population mean is not significantly more than 5 at αα = 0.01, so there is sufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is equal to 5.
   • The data suggest the populaton mean is significantly more than 5 at αα = 0.01, so there is sufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is more than 5.
8. Interpret the p-value in the context of the study.
   • There is a 72.5686636% chance of a Type I error.
   • If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds then there would be a 72.5686636% chance that the population mean number of cavities for twenty-year-olds would be greater than 5.
   • If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds then there would be a 72.5686636% chance that the sample mean for these 13 twenty-year-olds would be greater than 4.85.
   • There is a 72.5686636% chance that the population mean number of cavities for twenty-year-olds is greater than 5.
9. Interpret the level of significance in the context of the study.
   • If the population mean number of cavities for twenty-year-olds is more than 5 and if you survey another 13 twenty-year-olds, then there would be a 1% chance that we would end up falsely concuding that the population mean number of cavities for twenty-year-olds is equal to 5.
   • If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds, then there would be a 1% chance that we would end up falsely concuding that the population mean number of cavities for twenty-year-olds is more than 5.
   • There is a 1% chance that flossing will take care of the problem, so this study is not necessary.
   • There is a 1% chance that the population mean number of cavities for twenty-year-olds is more than 5.

Answer 1

(a) t-test for a population mean

(b) Ho: μ ≤ 5 and H1: μ > 5

(c)

n = 13   

μ = 5   

s = 0.9   

x-bar = 4.846   

SE = s/√n = 0.9/√13 = 0.249615088

t = (x-bar - μ)/SE = (4.846 - 5)/0.249615088301353 = -0.617

(d) p-value = 0.7256

(e) p-value > α

(f) Fail to reject Ho

(g)  The data suggest that the population mean number of cavities for twenty-year-olds is not significantly more than 5 at α = 0.01, so there is insufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is more than 5

(h) If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds then there would be a 72.5686636% chance that the sample mean for these 13 twenty-year-olds would be greater than 4.85

(i) If the population mean number of cavities for twenty-year-olds is 5 and if you survey another 13 twenty-year-olds, then there would be a 1% chance that we would end up falsely concluding that the population mean number of cavities for twenty-year-olds is more than 5.