Question

In the report "Healthy People 2020 Objectives for the Nation," The Centers for Disease Control and Prevention (CDC) set a goal of 0.341 for the proportion of mothers who will still be breastfeeding their babies one year after birth.† The CDC also estimated the proportion who were still being breastfed one year after birth to be 0.307 for babies born in 2013.† This estimate was based on a survey of women who had given birth in 2013. Suppose that the survey used a random sample of 1,000 mothers and that you want to use the survey data to decide if there is evidence that the goal is not being met. Let p denote the population proportion of all mothers of babies born in 2013 who were still breastfeeding at 12 months. (Hint: See Example 10.10. Use a table or technology.)

a)

Describe the shape, center, and variability of the sampling distribution of p̂ for random samples of size 1,000 if the null hypothesis H₀: p = 0.341 is true. (Round your standard deviation to four decimal places.)The shape of the sampling distribution is (left skewed / right skewed / approximately normal) . The sampling distribution is centered at

μ_p̂ =  .

The standard deviation of the sampling distribution is

σ_p̂ = .

B) Would you be surprised to observe a sample proportion as small as p̂ = 0.333 for a sample of size 1,000 if the null hypothesis H₀: p = 0.341 were true? Explain why or why not. (Round your answer to three decimal places.)I ( would /would not ) be surprised to observe a sample proportion of p̂ = 0.333for a sample of size 1,000 if the null hypothesis H₀: p = 0.341 is true. The probability of a sample proportion this small or smaller is __?___ which is (greater than / less than) the acceptance level of 0.05.

(c)Would you be surprised to observe a sample proportion as small as p̂ = 0.309 for a sample of size 1,000 if the null hypothesis H₀: p = 0.341 were true? Explain why or why not. (Round your answer to three decimal places.) I (would/ would not) be surprised to observe a sample proportion of p̂ = 0.309for a sample of size 1,000 if the null hypothesis H₀: p = 0.341 is true. The probability of a sample proportion this small or smaller is __?___ which is  (greater than / less than) the acceptance level of 0.05.

D) The actual sample proportion observed in the study was p̂ = 0.307.Based on this sample proportion, is there convincing evidence that the goal is not being met, or is the observed sample proportion consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation. (Round your answer to three decimal places.)

Since probability of a sample proportion this small or smaller is __?___ which is ( greater than / less than) the acceptance level of 0.05, there (is / is not) convincing evidence that the goal is not being met.

Answer 1

A)

The shape of the sampling distribution is approximately normal . The sampling distribution is centered at

μ_p̂ =  0.341

The standard deviation of the sampling distribution is

σ_p̂ = = 0.015

B)

Z score for p̂ = 0.333 is,

Z = (0.333 - 0.341) / 0.015 = -0.533

P(z < -0.533) = 0.297

Since p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis.

I would not be surprised to observe a sample proportion of p̂ = 0.333 for a sample of size 1,000 if the null hypothesis H₀: p = 0.341 is true. The probability of a sample proportion this small or smaller is __0.297___ which is greater than the acceptance level of 0.05.

C)

Z score for p̂ = 0.309 is,

Z = (0.309 - 0.341) / 0.015 = -2.133

P(z < -2.133) = 0.016

Since p-value is less than the significance level of 0.05, we reject the null hypothesis.

I would be surprised to observe a sample proportion of p̂ = 0.309 for a sample of size 1,000 if the null hypothesis H₀: p = 0.341 is true. The probability of a sample proportion this small or smaller is 0.016 which is less than the acceptance level of 0.05.

D)

Z score for p̂ = 0.307 is,

Z = (0.307 - 0.341) / 0.015 = -2.267

P(z < -2.267) = 0.012

For two-tail test, p-value = 2 * 0.012 = 0.024

Since p-value is less than the significance level of 0.05, we reject the null hypothesis.

Since probability of a sample proportion this small or smaller is __0.024___ which is less than the acceptance level of 0.05, there is convincing evidence that the goal is not being met.