Question

In: Statistics and Probability

As the national government has increased its voiced approval for the HPV vaccine known as Gardasil,...

As the national government has increased its voiced approval for the HPV vaccine known as Gardasil, many organizations have begun gathering data to address the rise in prevalence of young girls receiving the vaccine. Last year, the National Center for Health Statistics estimated that the National vaccine prevalence is up to 15% in the same age demographic. In order to test this assumption, a researcher conducted a study and found that out of 3,579 women in the demographic surveyed, 879 reported receiving one or more of the three shots included in the package for the vaccine.

A) Is the sample size large enough to justify the use of the Z formula?

B) Test if the proportion of the prevalence of the vaccine has changed. Use = 0.05. Hint: One sample proportion.

C) Calculate the 95% two-sided confidence interval for p and make a conclusion about H₀.

D) Compare your results and conclusions in a and b above. What would you conclude?

Carry probabilities to at least four decimal places for intermediate steps.

For extremely small probabilities, it is important to have one or two significant non-zero digits, for example, 0.000001 or 0.000034.

Round off your final answer to two decimal places.

*HAND Calculations ONLY, Show ALL steps*

Expert Solution

A) Since the sample size is larger than 30. It is enough to justify the use of the Z formula.

B)

H_o : The proportion of the prevalence of the vaccine has not changed.

H₁: The proportion of the prevalence of the vaccine has changed.

Test statistic = Z = ( p - p_o ) / ( p_o * ( 1 - p_o) / n )^0.5

We have ,

po = 0.15

p = 879 / 3579 = 0.2456

n = 3579

Test statistic = Z = ( 0.2456 - 0.15 ) / ( 0.15 * ( 1- 0.15) / 3579 )^0.5

Z = 16.017

P-value = P(Z > 16.017) = 0

Since, P-value = 0 < 0.05 (level of significance) , we reject Ho and conclude that the proportion of the prevalence of the vaccine has changed.

C) 95% two-sided confidence interval for p :

Upper limit = p + Z_0.05/2 * ( p_o * ( 1 - p_o) / n )^0.5 = 0.2456 + 1.96 * ( 0.15 * ( 1- 0.15 ) / 3579 )^0.5 = 0.2573

Lower limit = p - Z_0.05/2 * ( p_o * ( 1 - p_o) / n )^0.5 = 0.2456 - 1.96 * ( 0.15 * ( 1- 0.15 ) / 3579)^0.5

= 0.2339

95% two-sided confidence interval for p is ( 0.23 , 0.26) (rounded to 2 decimal place)

Since this confidence interval does not include 0.15, hence, Ho is rejected and we conclude that the proportion of the prevalence of the vaccine has changed.

D) We can see that , By testing the hypothesis and finding the confidence interval the same conclusion is obtained.

So it is true that the proportion of the prevalence of the vaccine has changed and it has increased.

orchestra answered 2 years ago

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