Question

From generation to generation, the mean age when smokers first start to smoke varies. However, the standard deviation of that age remains constant at around 2.1 years. A survey of 41 smokers of this generation was done to see if the mean starting age is at least 19. The sample mean was 18.1 with a sample standard deviation of 1.3. Do the data support the claim at the 5% level? Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Answer 1

`z=(18.1-19)/((2.1)/(√(41)))=-2.744`

`p_v =P(z<-2.744)=0.0030`

Since the p value is lower than the significance level we have enough evidence to reject the null hypothesis and we can say that the true mean is significantly lower than 19 and not support the statement provided that the mean starting age is at least 19

Explanation:

Data given and notation

`\bar X=18.1` represent the sample mean

`\sigma=2.1` represent the population standard deviation

`n=41` sample size

`\mu_o =19` represent the value that we want to test

`\alpha=0.05` represent the significance level for the hypothesis test.

z would represent the statistic (variable of interest)

`p_v` represent the p value for the test (variable of interest)

For this case we have the sample deviation but since the population deviation is also used we conduct the procedure with the population value.

System of hypothesis

We need to conduct a hypothesis in order to check if the true mean is at least 19, the system of hypothesis would be:

Null hypothesis:
`\mu \geq 19`

Alternative hypothesis:
`\mu < 19`

The statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

Calculate the statistic

We can replace in formula (1) the info given like this:

`z=(18.1-19)/((2.1)/(√(41)))=-2.744`

P-value

Since is a one side lower test the p value would be:

`p_v =P(z<-2.744)=0.0030`

Conclusion

Since the p value is lower than the significance level we have enough evidence to reject the null hypothesis and we can say that the true mean is significantly lower than 19 and not support the statement provided that the mean starting age is at least 19