To determine where most of the sample proportions fall, we can use the Central Limit Theorem to approximate the sampling distribution of sample proportions.
The mean of the sampling distribution of sample proportions is equal to the population proportion, which is 0.30 in this case. The standard deviation of the sampling distribution of sample proportions is calculated as follows:
σ_p = sqrt((p*(1-p))/n)
σ_p = sqrt((0.30*(1-0.30))/25)
σ_p = sqrt((0.21)/25)
σ_p = sqrt(0.0084)
σ_p = 0.0917
Using this standard deviation, we can calculate a one-standard deviation interval around the mean sample proportion of 0.30:
Lower bound: 0.30 - 0.0917 = 0.2083
Upper bound: 0.30 + 0.0917 = 0.3917
Thus, we can expect that approximately 68% of the sample proportions will fall within the interval of 0.2083 to 0.3917 when x = 0.30 and σ = 0.08.
Given 100 sample proportions of size 25, determine where most of the sample proportions fall when
x = 0.30 and σ = 0.08.
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