Question

You are interested in constructing a 99% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 405 randomly selected caterpillars observed, 45 lived to become butterflies. Round answers to 4 decimal places where possible. a. With 99% confidence the proportion of all caterpillars that lived to become a butterfly is between and b. If many groups of 405 randomly selected caterpillars were observed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about percent will not contain the true population proportion. Hint: Hints

Answer 1

Given that Sample size n = 405 X = 45 45 sample proportion X = = = n 0.1111 405 a) Now we have to construct the 99% confidence interval for true population proportion (p). Step (1): 99 31 - 1-0.99 = 0.01 100 0.01 0.005 2 Step (2): 1-0.005 = 0.995 Find the value of Za/2 such that P(Z < Za/2) = 0.995 From standard normal table Za/2 2.576 (Or) Use excel formula: =NORM.S.INV(0.995)

Step (3): Find the margin of error MOE. MOE = 2a/2 lô (1–0 0.1111*(1-0.1111) 2.576 * -0.0402 n 405 Step (4): Lower bound= p - MOE = 0.1111 – 0.0402 = 0.0709 Upper bound = P + MOE = 0.1111 + 0.0402 = 0.1513 :: The 99% confidence interval is between 0.0709 to 0.1513 b) ::About 99 percent of these confidence intervals will contain and about 1 percent will not contain.

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