Question

According to an article in Bloomberg Businessweek, New York City's most recent adult smoking rate is 14%. Suppose that a survey is conducted to determine this year's rate. Nine out of 70 randomly chosen N. Y. City residents reply that they smoke. Conduct a hypothesis test at the 5% level to determine if the rate has decreased from 14%.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 (c)

  In words, state what your random variable P' represents.
• a)P' represents the proportion of New York City residents who do not smoke.

• b)P' represents the proportion of New York City residents who smoke.

• c)P' represents the number of New York City residents who smoke.

• d) P' represents the average number of smokers in New York City.

• Part (d)

  State the distribution to use for the test. (Round your answers to four decimal places.)
  P' ~ ( , )

• Part (e)

  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.)

• Part (f)

  What is the p-value? (Round your answer to four decimal places.)
  
  
  (f2) Explain what the p-value means for this problem.
• a)If H₀ is true, then there is a chance equal to the p-value that the proportion of smokers in New York City is at least as different as the sample proportion is from 14%.
• b)If H₀ is false, then there is a chance equal to the p-value that the proportion of smokers in New York City is not at least as different as the sample proportion is from 14%.  
• c) If H₀ is true, then there is a chance equal to the p-value that the proportion of smokers in New York City is not at least as different as the sample proportion is from 14%.
• d)If H₀ is false, then there is a chance equal to the p-value that the proportion of smokers in New York City is at least as different as the sample proportion is from 14%.

• Part (g)

  Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value.

• 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 (Enter an exact number as an integer, fraction, or decimal.)
  α =
  
  (ii) Decision:

  reject the null hypothesis or do not reject the null hypothesis    

  
  (iii) Reason for decision:
• a)Since α < p-value, we do not reject the null hypothesis.

• b)Since α > p-value, we do not reject the null hypothesis.  

• c)Since α < p-value, we reject the null hypothesis.

• d)Since α > p-value, we reject the null hypothesis.

  
  (iv) Conclusion: a)There is sufficient evidence to support the claim that the percent of smokers in New York City is less than 14%. b)There is not sufficient evidence to support the claim that the percent of smokers in New York City is less than 14%.   

Answer 1

Part (c)

1) represents the proportion of New York City residents who smoke.

Part (d)

Part (e)

As sample size is greater than 30. So, we will use z-test for proportion.

Favourable number of cases who smoke are 9.

• What is the test statistic.  

• Part (f) :- The p-value is 0.3914.

(f2) P-value represents the probability of obtaining the test resuls as extreme as from the result considering that the null hypothesis is true.

Part (g)

b)There is not sufficient evidence to support the claim that the percent of smokers in New York City is less than 14%.

Part (h)

a)Since p-value > α , we do not reject the null hypothesis.  

If H0 is true, then there is a chance equal to the p-value that the proportion of smokers in New York City is at least as different as the sample proportion is from 14%.