A psychologist is interested in constructing a $90 \%$ confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 68 of the 815 randomly selected people who were surveyed agreed with this theory.
a. With $90 \%$ confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between and
b. If many groups of 815 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about percent will not contain the true population proportion.
Answer
So, with 90% confidence, the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between 0.0675 and 0.0994. Therefore, the final answer is \(\boxed{[0.0675, 0.0994]}\).
Step 1 ：First, we calculate the sample proportion (p̂) by dividing the number of people who agreed with the theory by the total number of people surveyed. In this case, p̂ = \(\frac{68}{815}\) = 0.0834355828220859.
Step 2 ：Next, we find the Z-score corresponding to the desired confidence level. For a 90% confidence level, the Z-score (Z) is 1.645.
Step 3 ：We then calculate the standard error (se) using the formula \(\sqrt{\frac{p̂(1-p̂)}{n}}\), where n is the sample size. Substituting the values we have, se = \(\sqrt{\frac{0.0834355828220859(1-0.0834355828220859)}{815}}\) = 0.009686755637183632.
Step 4 ：Finally, we calculate the confidence interval using the formula p̂ ± Z*se. The lower limit of the confidence interval (ci_lower) is 0.0834355828220859 - 1.645*0.009686755637183632 = 0.06750086979891881, and the upper limit (ci_upper) is 0.0834355828220859 + 1.645*0.009686755637183632 = 0.09937029584525298.
Step 5 ：So, with 90% confidence, the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between 0.0675 and 0.0994. Therefore, the final answer is \(\boxed{[0.0675, 0.0994]}\).
