Question

The average number of cavities that thirty-year-old Americans have had in their lifetimes is 7. Do twenty-year-olds have fewer cavities? The data show the results of a survey of 12 twenty-year-olds who were asked how many cavities they have had. Assume that the distribution of the population is normal.

4, 5, 6, 6, 8, 6, 7, 4, 8, 5, 4, 7

What can be concluded at the αα = 0.05 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  Select an answer < > ≠ =       

H1:H1:  ? p μ  Select an answer > = < ≠    

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 less than 7 at αα = 0.05, so there is insufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is less than 7.
   • The data suggest the populaton mean is significantly less than 7 at αα = 0.05, so there is sufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is less than 7.
   • The data suggest the population mean is not significantly less than 7 at αα = 0.05, so there is sufficient evidence to conclude that the population mean number of cavities for twenty-year-olds is equal to 7.
8. Interpret the p-value in the context of the study.
   • There is a 0.93566304% chance that the population mean number of cavities for twenty-year-olds is less than 7.
   • There is a 0.93566304% chance of a Type I error.
   • If the population mean number of cavities for twenty-year-olds is 7 and if you survey another 12 twenty-year-olds, then there would be a 0.93566304% chance that the sample mean for these 12 twenty-year-olds would be less than 5.83.
   • If the population mean number of cavities for twenty-year-olds is 7 and if you survey another 12 twenty-year-olds, then there would be a 0.93566304% chance that the population mean number of cavities for twenty-year-olds would be less than 7.
9. Interpret the level of significance in the context of the study.
   • If the population mean number of cavities for twenty-year-olds is 7 and if you survey another 12 twenty-year-olds, then there would be a 5% chance that we would end up falsely concuding that the population mean number of cavities for twenty-year-olds is less than 7.
   • If the population mean number of cavities for twenty-year-olds is less than 7 and if you survey another 12 twenty-year-olds, then there would be a 5% chance that we would end up falsely concuding that the population mean number of cavities for twenty-year-olds is equal to 7.
   • There is a 5% chance that the population mean number of cavities for twenty-year-olds is less than 7.
   • There is a 5% chance that flossing will take care of the problem, so this study is not necessary.

Answer 1

solution:

first of all calculating the mean and standard deviation of the sampple

mean =

standard deviation;

a) since population standard deviation is not known so we will use t test

b) the null and alternative hpothesis:

c) test statistics:

d) = 0.05

degree of freedom = 12-1 = 11

p value = 0.0093

p value < = 0.05

d) so   we are reject the null hypothesis.

e) the data suggest that the population mean is less than 7 at 0.05, so there is sufficient evidence to concude that the population mean number of cavities for twenty year old is less than 7