A researcher wants to estimate the proportion of depressed individuals taking a new anti-depressant drug who find relief. A random sample of 275 individuals who had been taking the drug is questioned; 214 of them found relief from depression. Based upon this, compute a 95% confidence interval for the proportion of all depressed individuals taking the drug who find relief. Then find the lower limit and upper limit of the 95% confidence interval.Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. (If necessary, consult a list of formulas.)Lower limit: Upper limit:
Question
A researcher wants to estimate the proportion of depressed individuals taking a new anti-depressant drug who find relief. A random sample of 275 individuals who had been taking the drug is questioned; 214 of them found relief from depression. Based upon this, compute a 95% confidence interval for the proportion of all depressed individuals taking the drug who find relief. Then find the lower limit and upper limit of the 95% confidence interval.Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. (If necessary, consult a list of formulas.)Lower limit: Upper limit:
Solution
To calculate the 95% confidence interval for the proportion of all depressed individuals taking the drug who find relief, we first need to find the sample proportion (p̂) and the standard error (SE).
Step 1: Calculate the sample proportion (p̂) The sample proportion (p̂) is calculated as the number of individuals who found relief (214) divided by the total number of individuals in the sample (275).
p̂ = 214 / 275 = 0.778
Step 2: Calculate the standard error (SE) The standard error (SE) is calculated using the formula: SE = sqrt [ p̂(1 - p̂) / n ] where n is the sample size.
SE = sqrt [ 0.778(1 - 0.778) / 275 ] = 0.026
Step 3: Calculate the 95% confidence interval The 95% confidence interval is calculated using the formula: p̂ ± Z*SE where Z is the Z-value from the standard normal distribution corresponding to the desired confidence level (for a 95% confidence level, Z = 1.96).
Lower limit = p̂ - ZSE = 0.778 - 1.960.026 = 0.727 Upper limit = p̂ + ZSE = 0.778 + 1.960.026 = 0.829
So, the 95% confidence interval for the proportion of all depressed individuals taking the drug who find relief is (0.727, 0.829). This means we are 95% confident that the true proportion of all depressed individuals taking the drug who find relief is between 72.7% and 82.9%.
Lower limit: 0.73 Upper limit: 0.83
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