Question

Researchers interviewed street prostitutes in Canada and the United States. The mean age of the 100 Canadian prostitutes upon entering prostitution was 18 with a standard deviation of six. The mean age of the 130 United States prostitutes upon entering prostitution was 20 with a standard deviation of eight. Is the mean age of entering prostitution in Canada lower than the mean age in the United States? Test at a 1% significance level.

NOTE: If you are using a Student's t-distribution for the problem, including for paired data, 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: μ_C ≠ μ_US

  H₀: μ_C = μ_US

      

  H₀: μ_C < μ_US

  H₀: μ_C > μ_US

• Part (b)

  State the alternative hypothesis.

  H_a: μ_C > μ_US

  H_a: μ_C = μ_US

      

  H_a: μ_C ≠ μ_US

  H_a: μ_C < μ_US

• Part (c)

  In words, state what your random variable

  X_C − X_US

  represents.

  X_C − X_US

  represents the mean difference in the age of entering prostitution in Canada and the United States.

  X_C − X_US

  represents the mean age of entering prostitution in Canada and the United States.    

  X_C − X_US

  represents the difference in the ages of entering prostitution in Canada and the United States.

  X_C − X_US

  represents the difference in the mean age of entering prostitution in Canada and the United States.

• Part (d)

  State the distribution to use for the test. (Enter your answer in the form z or t_df where df is the degrees of freedom. Round your answer to two decimal places.)

• Part (e)

  What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)
  ---Select--- z t =

• Part (f)

  What is the p-value?

  p-value < 0.0100.010 < p-value < 0.050    0.050 < p-value < 0.100p-value > 0.100

  
  
  Explain what the p-value means for this problem.If

  H₀

  is false, then there is a chance equal to the p-value that the sample mean age of entering prostitution in Canada is at least 2 more than the sample mean age of entering prostitution in the United States.If

  H₀

  is true, then there is a chance equal to the p-value that the sample mean age of entering prostitution in Canada is at least 2 less than the sample mean age of entering prostitution in the United States.    If

  H₀

  is true, then there is a chance equal to the p-value that the sample mean age of entering prostitution in Canada is at least 2 more than the sample mean age of entering prostitution in the United States.If

  H₀

  is false, then there is a chance equal to the p-value that the sample mean age of entering prostitution in Canada is at least 2 less than the sample mean age of entering prostitution in the United States.

• 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 hypothesisdo not reject the null hypothesis    

  
  (iii) Reason for decision:

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

  
  (iv) Conclusion:

  There is sufficient evidence to show that the mean age of entering prostitution in Canada is lower than the mean age in the United States.There is not sufficient evidence to show that the mean age of entering prostitution in Canada is lower than the mean age in the United States.    

• Part (i)

  Explain how you determined which distribution to use.

  The standard normal distribution will be used because the samples involve the difference in proportions.The t-distribution will be used because the samples are dependent.    The standard normal distribution will be used because the samples are independent and the population standard deviation is known.The t-distribution will be used because the samples are independent and the population standard deviation is not known.

Answer 1

Part (a)

State the null hypothesis

H₀: μ_C = μ_US

Part (b)

State the alternative hypothesis.

H_a: μ_C < μ_US

Part (c)

In words, state what your random variable

X_C − X_US

represents

X_C − X_US

represents the mean difference in the age of entering prostitution in Canada and the United States.

Part (d)

State the distribution to use for the test.

t -distribution with df = 228

Part (e)

What is the test statistic?

Part (f)

What is the p-value?

P-value =0.0189

Part (g)

Sketch a picture of this situation.

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 = 0.01

(ii) Decision:

do not reject the null hypothesis .

(iii) Reason for decision:

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

(iv) Conclusion:

There is not sufficient evidence to show that the mean age of entering prostitution in Canada is lower than the mean age in the United States.

Part (i)

Explain how you determined which distribution to use.

The t-distribution will be used because the samples are independent and the population standard deviation is not known.

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