A sample of size 300 is to be taken at random from an infinite population. Given that the population proportion is 0.70, the probability that the sample proportion will be smaller than 0.75 is about: Group of answer choices 0.53. 0.47. 0.03. 0.97.
Question
A sample of size 300 is to be taken at random from an infinite population. Given that the population proportion is 0.70, the probability that the sample proportion will be smaller than 0.75 is about: Group of answer choices
0.53.
0.47.
0.03.
0.97.
Solution
To solve this problem, we need to use the concept of sampling distribution of sample proportions.
Step 1: Identify the given values. The population proportion (p) is 0.70, the sample size (n) is 300, and we want to find the probability that the sample proportion (p̂) is less than 0.75.
Step 2: Calculate the standard deviation of the sampling distribution (σp̂). The formula for this is sqrt[(p(1 - p) / n], which is sqrt[(0.70 * 0.30) / 300] = 0.027.
Step 3: Find the z-score for the sample proportion 0.75. The formula for the z-score is (p̂ - p) / σp̂, which is (0.75 - 0.70) / 0.027 = 1.85.
Step 4: Look up the z-score in the z-table to find the probability. The value for 1.85 in the z-table is 0.9671, which means the probability that the sample proportion is less than 0.75 is about 0.97.
So, the correct answer is 0.97.
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