The marketing manager for the print division of Publishing and Broadcasting Limited claims that 23% of university students regularly read the Bulletin magazine. A survey of 240 students showed that only 43 students read the Bulletin regularly. Assuming the manager's claim is correct, determine (to 4 decimal places):1. the standard error for the sampling distribution of the proportion. 2. the probability that the sample proportion is no more than that found in the survey.
Question
The marketing manager for the print division of Publishing and Broadcasting Limited claims that 23% of university students regularly read the Bulletin magazine. A survey of 240 students showed that only 43 students read the Bulletin regularly. Assuming the manager's claim is correct, determine (to 4 decimal places):1. the standard error for the sampling distribution of the proportion. 2. the probability that the sample proportion is no more than that found in the survey.
Solution
- The standard error for the sampling distribution of the proportion is calculated using the formula:
SE = sqrt[p(1 - p) / n]
where p is the proportion (0.23 in this case) and n is the sample size (240 in this case).
SE = sqrt[0.23(1 - 0.23) / 240] = 0.0283 (rounded to four decimal places)
- To find the probability that the sample proportion is no more than that found in the survey, we first need to standardize the sample proportion using the z-score formula:
Z = (p̂ - p) / SE
where p̂ is the sample proportion (43 / 240 = 0.1792) and p is the proportion (0.23).
Z = (0.1792 - 0.23) / 0.0283 = -1.798 (rounded to three decimal places)
We then look up this z-score in a standard normal distribution table or use a calculator with a normal distribution function to find the probability. The probability that Z is less than -1.798 is 0.0362 or 3.62%.
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