Question

According to an article in Newsweek, the natural ratio of girls to boys is 100:105. In China, the birth ratio is 100:114 (46.7% girls). Suppose you don't believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 62 girls and 88 boys born of the 150. Based on your study, do you believe that the percent of girls born in China is 46.7? Conduct a hypothesis test 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:

  p ≠ 0.467

  H₀:

  p ≥ 0.467    

  H₀:

  p = 0.467

  H₀:

  p ≤ 0.467

• Part (b)

  State the alternative hypothesis.

  H_a:

  p = 0.467

  H_a:

  p > 0.467    

  H_a:

  p < 0.467

  H_a:

  p ≠ 0.467

• Part (c)

  In words, state what your random variable P' represents.

  P' represents the percent of boys born in China.P' represents the percent of girls born in China.    P' represents the number of girls born in China.P' represents the ratio of girls to boys in China.

• 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.)
  ---Select--- z t =

• 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 sample ratio is not 62 out of 150 or less OR 78 out of 150 or more.If

  H₀

  is false, then there is a chance equal to the p-value that the sample ratio is not 62 out of 150 or less OR 78 out of 150 or more.    If

  H₀

  is true, then there is a chance equal to the p-value that the sample ratio is 62 out of 150 or less OR 78 out of 150 or more.If

  H₀

  is false, then there is a chance equal to the p-value that the sample ratio is 62 out of 150 or less OR 78 out of 150 or more.

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

  
  (iv) Conclusion:

  There is sufficient evidence to conclude that the percent of girls born in China is not equal to 46.7%.There is not sufficient evidence to conclude that the percent of girls born in China is not equal to 46.7%.    

• Part (i)

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

Answer 1

Part (a)

the null hypothesis.

H₀:

p = 0.467

Part (b)

the alternative hypothesis

H_a:

p ≠ 0.467

Part (c)

P' represents the percent of girls born in China

Part (d)

The distribution to use for the test. (Round your answers to four decimal places.)
P' ~ Normal( mu= 0.467, sigma= 0.2489)

Part (e)

The test statistic? (If using the z distribution round your answers to two decimal places
z = (0.413-0.467) / sqrt( ( 0.467*0.533)/150)

z= -1.33

Part (f)

P-value = P(|Z|> 1.33) = 0.183518

The P-Value is 0.1835.

p-value means for this problem

if H₀

is true, then there is a chance equal to the p-value that the sample ratio is 62 out of 150 or less OR 78 out of 150 or more

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)

i) alpha= 0.005

ii) do not reject the null hypothesis   

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

(iv) Conclusion: There is not sufficient evidence to conclude that the percent of girls born in China is not equal to 46.7%.  

Part(i) Construct a 95% confidence interval for the true proportion is

( p' - 1.96 * sqrt( p' * (1-p')/n) , p' + 1.96 * sqrt( p' * (1-p')/n))

( 0.413 - 1.96 * sqrt( 0.413* 0.587/ 150) , 0.4133 + 1.96 * sqrt( 0.4133* 0.5877/ 150))

( 0.4133 - 0.07880 , 0.4133 - 0.07880

( 0.3345, 0.4921)

95% confidence interval for the true proportion is ( 0.3345, 0.4921) .