Step 1 :Given that 23% of a sample of 400 are in Category A, we can calculate the point estimate for the proportion p as 0.23.
Step 2 :Next, we calculate the standard error of the proportion using the formula \(\sqrt{p(1-p)/n}\), where p is the point estimate and n is the sample size. Substituting the given values, we get the standard error as approximately 0.021.
Step 3 :We then calculate the margin of error by multiplying the standard error by the z-score for the desired level of confidence. For a 90% confidence level, the z-score is approximately 1.645. This gives us a margin of error of approximately \(\pm 0.035\).
Step 4 :Finally, we calculate the 90% confidence interval by adding and subtracting the margin of error from the point estimate. This gives us a confidence interval of \((0.195, 0.265)\).
Step 5 :\(\boxed{\text{The point estimate for } p \text{ is } 0.23. \text{ The margin of error is } \pm 0.035. \text{ The 90% confidence interval is } (0.195, 0.265)}\)