From generation to generation, the average age when smokers first start to smoke

Question

From generation to generation, the average 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 average starting age is at least 19. The sample average 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.)

Part (a) State the null hypothesis.

H0: ? < 19

H0: ? = 19   

H0: ? ? 19

H0: ? ? 19

Part (b) State the alternative hypothesis.

Ha: ? ? 19

Ha: ? = 19   

Ha: ? ? 19

Ha: ? < 19

Part (c) In words, state what your random variable X"bar" represents.

X"bar" represents the average number of cigarettes for each smoker.

X"bar" represents the age of a person when he or she first began smoking.   

X"bar" represents the average age of the sample of smokers.

X"bar" represents the average age when smokers first start to smoke.

Part (d) State the distribution to use for the test. (Round your standard deviation to four decimal places.)

X"bar" ~ ( , )

Part (e) What is the test statistic? (Round your answer to two decimal places.)

Select (t or z) =

Part (f) What is the p-value? (Round your answer to four decimal places.)

( )

Explain what the p-value means for this problem.

If H0 is true, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is not 18.1 years or less.

If H0 is false, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is not 18.1 years or less.

If H0 is true, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is 18.1 years or less.

If H0 is false, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is 18.1 years or less.

Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. (Upload your file below.)

Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.

(i) Alpha:

? =  

(ii) Decision:

reject the null hypothesis.

do not reject the null hypothesis.

(iii) Reason for decision:

Since ? < p-value, we do not reject the null hypothesis.

Since ? < p-value, we reject the null hypothesis.     

Since ? > p-value, we reject the null hypothesis.

Since ? > p-value, we do not reject the null hypothesis.

(iv) Conclusion:

There is sufficient evidence to conclude that the starting age for smoking in this generation is lower than 19.

There is not sufficient evidence to conclude that the starting age for smoking in this generation is lower than 19.     

Part (i) Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your lower and upper bounds to two decimal places.)

Explanation / Answer

Part (A)
Correct answer: option (B) H0: Mu = 19
Part (B)
Correct answer: option (A) H0: Mu > 19

Part (C) X"bar" represents the average age of the sample of smokers.

Part (D) Xbar follows N(18.1,2.1)

Part (E) Z Statistic = (18.1 - 19) / (2.1/sqrt(41)) = -2.7442

Part (F) P-Value: 0.9970
Correct answer: If H0 is true, then there is a chance equal to the p-value that the average age of people when they first begin to smoke is not 18.1 years or less.

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