Question

The U.S. Federal Seed Act establishes germination rates for various fruit and vegetable seeds. Watermelon seeds are to meet a 70% germination standard. A skeptical gardener who has not had very good luck planting watermelons believes that the seed company he purchases seeds from is not adhering to the 70% federal mandate. Once a week for 12 weeks, he purchases a pack of 10 watermelon seeds to act as his sample. He plants the seeds in a greenhouse with good soil to maintain a consistent temperature and watering routine. He finds that the germination rate for the company's watermelon seeds is 65%. Is this convincing evidence that the proportion of watermelon seeds that germinate is less than 70%?

(a)

Compute a 95% confidence interval to estimate the proportion of watermelon seeds that germinate. Interpret your interval. (Round your answers to three decimal places.)

Based upon this sample, we can be 95% confident that the true proportion of watermelon seeds that germinate is between a lower value of _____ and a upper value of _____

Answer 1

sample proportion, = 0.65
sample size, n = 10
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.65 * (1 - 0.65)/10) = 0.1508

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.1508
ME = 0.2956

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.65 - 1.96 * 0.1508 , 0.65 + 1.96 * 0.1508)
CI = (0.354 , 0.946)

Based upon this sample, we can be 95% confident that the true proportion of watermelon seeds that germinate is between a lower value of 0.354 and a upper value of 0.946