Final answer:
The probability that the sample proportion will be less than 0.07, given that the population proportion is 0.06 and the sample size is 300, is approximately 0.7580 when rounded to four decimal places.
Step-by-step explanation:
The student is asking about the probability that the sample proportion will be less than 0.07 when the true population proportion is 0.06. Since the sample size is 300, we know by the Central Limit Theorem that the sampling distribution of the sample proportions will be approximately normally distributed if certain conditions are met. Given that the population proportion is 0.06 and assuming np and n(1-p) are both greater than 5, this condition is satisfied for normal approximation.
To find this probability, we first need to determine the mean (μ_p) and standard deviation (σ_p) of the sampling distribution. The mean of the sampling distribution (μ_p) is equal to the population proportion, which is 0.06, and the standard deviation (σ_p) is calculated as the square root of [(p(1 - p)) / n].
Calculating standard deviation: σ_p = √[(0.06)(0.94) / 300] = 0.0143
Next, we use Z-scores to find the probability. The Z-score is calculated as (X - μ_p) / σ_p, where X is the sample proportion for which we want to find the probability. In this case, X is 0.07.
Calculating Z-score: Z = (0.07 - 0.06) / 0.0143 ≈ 0.6993
Using the Z-table or a calculator that provides normal distribution probabilities, we can find the probability that Z is less than 0.6993.
The probability obtained from the Z-table is approximately 0.7580, which means there's about a 75.80% chance that the sample proportion will be less than 0.07.
Therefore, the probability that the sample proportion is less than 0.07 is 0.7580 when rounded to four decimal places.