The weights of newborn children in the United States vary according to the normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. An article in the The Wall Street Journal claims that the average birth weight is decreasing over time. Steady Decline Change in the average birth weight of bables" born in the US. since 1990 , in ounces Bert betave 57 and 41 weis and nat noluding muliple tirths Sturioe Obrtaties and Ginacitoy A local hospital wanted to see if the claim was true and randomly sampled 21 baby's weights and found an average weight of 7.4 pounds. Test the claim using a $1 \%$ level of significance that the mean weight significantly smatter than the national average of 7.5 pounds. Use the poputation standard deviation of 1.25 pounds. Give answer to at least 4 decimal places. What are the correct hypotheses? (Select the correct symbols.) $\mathrm{H}_{0}:$ Select an answerv ? pounds $\mathrm{H}_{1}:$ Select an answerv ? pounds Based on the hypotheses, find the following: Test Statistic $=$ \[ p \text {-value }= \] The correct decision is to Select an answer
Solution
Step 1 ：Given values are: sample size = 21, sample mean = 7.4 pounds, population mean = 7.5 pounds, population standard deviation = 1.25 pounds, and significance level = 0.01.
Step 2 ：The null hypothesis \(H_0\) is that the population mean \(\mu\) is equal to 7.5 pounds, i.e., \(H_0: \mu = 7.5\).
Step 3 ：The alternative hypothesis \(H_1\) is that the population mean \(\mu\) is less than 7.5 pounds, i.e., \(H_1: \mu < 7.5\).
Step 4 ：Calculate the test statistic (z) using the formula: \(z = \frac{{\text{{sample mean}} - \text{{population mean}}}}{{\text{{population standard deviation}} / \sqrt{{\text{{sample size}}}}}}\).
Step 5 ：Substitute the given values into the formula to get: \(z = \frac{{7.4 - 7.5}}{{1.25 / \sqrt{21}}}\), which gives \(z \approx -0.3666\).
Step 6 ：Calculate the p-value using the cumulative distribution function (CDF) for the normal distribution. The p-value is the probability that a random variable is less than or equal to the test statistic, which gives \(p \approx 0.3570\).
Step 7 ：Since the p-value (0.3570) is greater than the significance level (0.01), we fail to reject the null hypothesis.
Step 8 ：\(\boxed{\text{{Therefore, we do not have enough evidence to support the claim that the average birth weight is decreasing over time.}}}\)
