Based on their records, a hospital claims that the proportion, p, of full-term babies born in the community that weigh more than 7 pounds is 44%. A pediatrician who works with several hospitals in the community would like to verify the hospital's claim. In a random sample of 185 babies born in the community, 77 weighed over 7 pounds. Is there enough evidence to reject the hospital's claim at the 0.01 level of significance? Perform a two-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.) (a) State the null hypothesis H_0 and the alternative hypothesis H_1. (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the p-value. (Round to three or more decimal places.) (e) Can we reject the claim that the proportion of full-term babies born in the community that weigh more than 7 pounds is 44%?
Question
Based on their records, a hospital claims that the proportion, p, of full-term babies born in the community that weigh more than 7 pounds is 44%. A pediatrician who works with several hospitals in the community would like to verify the hospital's claim. In a random sample of 185 babies born in the community, 77 weighed over 7 pounds. Is there enough evidence to reject the hospital's claim at the 0.01 level of significance?
Perform a two-tailed test. Then complete the parts below.
Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.) (a) State the null hypothesis H_0 and the alternative hypothesis H_1. (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the p-value. (Round to three or more decimal places.) (e) Can we reject the claim that the proportion of full-term babies born in the community that weigh more than 7 pounds is 44%?
Solution
(a) The null hypothesis H_0: p = 0.44 (The proportion of full-term babies born in the community that weigh more than 7 pounds is 44%) The alternative hypothesis H_1: p ≠ 0.44 (The proportion of full-term babies born in the community that weigh more than 7 pounds is not 44%)
(b) The type of test statistic to use is a z-test for proportions, because we are comparing a sample proportion to a claimed population proportion.
(c) To find the value of the test statistic, we first need to calculate the sample proportion (p̂) which is 77/185 = 0.416. Then we use the formula for the z-test statistic: z = (p̂ - p) / sqrt[(p(1 - p) / n]. Substituting the given values, we get z = (0.416 - 0.44) / sqrt[(0.44 * 0.56) / 185] = -0.024 / 0.034 = -0.706 (rounded to three decimal places).
(d) The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. We can find this value using a standard normal (Z) distribution table or a calculator with a p-value function. For a two-tailed test, we find the probability that Z is less than -0.706 or greater than 0.706. This gives us a p-value of 0.480 (rounded to three decimal places).
(e) Since the p-value (0.480) is greater than the level of significance (0.01), we do not reject the null hypothesis. Therefore, there is not enough evidence to reject the hospital's claim that the proportion of full-term babies born in the community that weigh more than 7 pounds is 44%.
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