From generation to generation, the mean age when smokers first start to smoke va

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 18.2 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.)

1. State the distribution to use for the test. (Round your answers to four decimal places.)
X ~
2. What is the test statistic? (If using the z distribution round your answers to two decimal places, and if using the t distribution round your answers to three decimal places.)
3. What is the p-value? (Round your answer to four decimal places.)
4. (i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
(ii) Decision:
(iii) Reason for decision:
(iv) Conclusion:
5. Construct a 95% confidence interval for the true mean.

Explanation / Answer

1) we will use normal distribution and one sample z test as population stdd deviation is given

2)null hypothesis: mean age =19

alternate hypothesis: mean age<19

here std error =std deviation/(n)1/2=0.328

hence test stat z=(X-mean)/std error =(18.2-19)/0.328=-2.44

for above p value=0.007

4)(i)alpha=0.05

(ii) reject the null hypothesis

(iii) as p value is less then 0.05 level

(iv) mean age is less then 19 years

5) for 95% CI, z=1.96

hence confidence interval =mean +/- z*std deviation =17.557 ; 18.843

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