A market survey was conducted to estimate the proportion of homemakers who could recognize the brand name of a cleanser based on the shape and color of the container. Of the 1,400 homemakers, 420 were able to identify the brand name. Using the 0.99 degree of confidence, the population proportion lies within what interval?
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A. B. C. D. E.B
Interval estimate can be found from p +/- z[p(1-p)/n]^0.5. Here we have n = 1400, p = 414/1400 = 0.3 and z = 2.58 (for 99%).
Therefore 0.3 +/- 2.58*0.01225 and we get 0.268 and 0.332.
To determine the interval within which the population proportion lies, we can use the confidence interval formula. The formula for the confidence interval of a proportion is:
CI=p^±Z×np^(1−p^)
where:
In this case, we are given that the sample size (n) is 1,400 and the sample proportion (p^) is 420/1,400 = 0.3 (since 420 homemakers out of 1,400 were able to identify the brand name).
To find the z-score corresponding to a 99% confidence level, we can refer to the standard normal distribution table or use a statistical software. The z-score for a 99% confidence level is approximately 2.576.
Now we can substitute the values into the formula to calculate the confidence interval:
CI=0.3±2.576×1,4000.3(1−0.3)
Simplifying the equation:
CI=0.3±2.576×1,4000.3×0.7
CI=0.3±2.576×1,4000.21
CI=0.3±2.576×0.00015
Calculating the square root:
CI=0.3±2.576×0.0122474
CI=0.3±0.031546
Therefore, the confidence interval for the population proportion lies between 0.268 (0.3 - 0.031546) and 0.332 (0.3 + 0.031546).
The answer is option B: 0.268 and 0.332.