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The article "Breast-Feeding Rates Up Early" (USA Today, Sept. 14,2010 ) summarizes a survey of mothers whose babies were born in \(2009 .\) The Center for Disease Control sets goals for the proportion of mothers who will still be breast-feeding their babies at various ages. The goal for 12 months after birth is 0.25 or more. Suppose that the survey used a random sample of 1,200 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 proportion of all mothers of babies born in 2009 who were still breast-feeding at 12 months. (Hint: See Example 10.10 ) a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for random samples of size 1,200 if the null hypothesis \(H_{0}: p=0.25\) is true. b. Would you be surprised to observe a sample proportion as small as \(\hat{p}=0.24\) for a sample of size 1,200 if the null hypothesis \(H_{0}: p=0.25\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as small as \(\hat{p}=0.20\) for a sample of size 1,200 if the null hypothesis \(H_{0}: p=0.25\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.22 .\) 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.

Step by step solution

01

Calculation of Standard Deviation

The shape of the sampling distribution will be approximately normal due to the large sample size (n = 1,200). The center of the distribution will be at \(p = 0.25\) (mean = \(p\)). The spread of the sampling distribution is characterized by its standard deviation, which we can calculate using the formula \(\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}\), where \(p\) is the proportion under the null hypothesis and \(n = 1200\). The standard deviation, \(\sigma_{\hat{p}}\) is therefore, \(\sqrt{0.25(1 - 0.25)/1200} \approx 0.0128\).

02

Would you be surprised if \(\hat{p} = 0.24\)?

Next, we calculate the Z-score for the sample proportion \(0.24\) using the formula \(Z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}\). Setting \(\hat{p} = 0.24\), we get \(Z = (0.24 - 0.25)/0.0128 = -0.78\). As the z-score is within -2 to 2, the sample proportion of \(0.24\) is not surprising given the null hypothesis of \(0.25\).

03

Would you be surprised if \(\hat{p} = 0.20\)?

We repeat the previous step with \(\hat{p} = 0.20\). The z-score we get is \(Z = (0.20 - 0.25)/0.0128 = -3.91\). This z-score lies outside the range of -2 to 2, therefore, we would be surprised to observe a sample proportion as small as \(0.20\). This suggests that if the true proportion were \(0.25\), it would be rare to observe a sample proportion of \(0.20\).

04

The sample proportion observed in the study

Finally, we need to calculate the Z-score for the sample proportion from the study, \(\hat{p} = 0.22\). This z-score is \(Z = (0.22 - 0.25)/0.0128 = -2.34\). Since this Z-score falls outside the range of -2 to 2 (though it's close), it provides some evidence against the null hypothesis. Although it's not too surprising to have this sample proportion if the goal is met, it's more likely to be observed when the actual population proportion is less than \(0.25\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sampling Distribution

When conducting a hypothesis test, the concept of a sampling distribution is crucial. Imagine you took many random samples from a population and calculated a statistic, like the sample proportion, for each of those samples. The sampling distribution is the distribution of those sample statistics.

In our case, we are looking at the sampling distribution of the sample proportion, \(\hat{p}\), with the null hypothesis \(H_0: p = 0.25\). The beautiful thing about the sampling distribution is that it is approximately normal, especially with large sample sizes like \(n = 1,200\). The center of this normal distribution is our hypothesized population proportion, \(p = 0.25\).

The spread of this distribution is determined by its standard deviation, \(\sigma_{\hat{p}}\), which we computed as \(\sqrt{\frac{0.25(1-0.25)}{1200}} \approx 0.0128\). This gives us an idea of how much individual sample proportions will vary from the expected population proportion.

Statistical Significance

Statistical significance helps us determine whether the results of a hypothesis test are meaningful or could just be due to random chance. When we calculate a Z-score, we're trying to find out how far our sample proportion, \(\hat{p}\), deviates from the expected population proportion under the null hypothesis.

If this deviation is large enough, it's considered statistically significant. In simpler terms, the result is not easily explained by random chance alone. Typically, a Z-score falling outside the range of -2 to 2 suggests statistical significance, indicating that our sample result is less likely to occur if the null hypothesis is true.

Z-Score

A Z-score is a numerical measurement used in hypothesis testing to describe a value's relationship to the mean of a group of values. It's calculated by:

• Subtracting the population mean (expected value) from the observed value
• Dividing the result by the standard deviation

In the context of our breast-feeding survey, a Z-score helps to understand how far our sample proportion is from the hypothesized 0.25.

For example, when \(\hat{p} = 0.24\), the Z-score is calculated as \(Z = \frac{0.24 - 0.25}{0.0128} = -0.78\). A Z-score of -0.78 indicates the sample proportion is close to the mean and not statistically surprising. Conversely, \(\hat{p} = 0.20\) gives a Z-score of \(-3.91\), showing a substantial deviation from the mean, implying it's much less likely to occur if the null hypothesis is true.

Null Hypothesis

In hypothesis testing, the null hypothesis, denoted as \(H_0\), serves as the baseline or default assumption. It proposes that there is no significant effect or relationship, which in this scenario means the proportion of mothers still breast-feeding is truly 0.25.

The null hypothesis is what we test against. We collect data and calculate probabilities to see whether evidence is strong enough to reject it. If our sample data significantly deviates from what the null hypothesis predicts, we might claim evidence against it. However, rejecting the null doesn't prove it false; it merely suggests it's unlikely given the observed data.

Probability Calculations

Probability calculations allow us to quantify the likelihood of observing our sample result under the null hypothesis. This is essential in making informed decisions during hypothesis testing.

By calculating a Z-score and finding the corresponding probability, we can see how extreme our sample result is compared to the null hypothesis. For instance, with our observed sample proportion \(\hat{p} = 0.22\), the Z-score is \(Z = -2.34\). This corresponds to a small probability of observing such or more extreme results under \(H_0: p = 0.25\), suggesting the observed deviation is somewhat surprising. This supports a view that the true population proportion is possibly less than 0.25.