A music industry researcher wants to estimate, with a 90% confidence level, the proportion of young urban people (ages 21 to 35 years) who go to at least 3 concerts a year. Previous studies show that 21% of those people (21 to 35 year olds) interviewed go to at least 3 concerts a year. The researcher wants to be accurate within 1% of the true proportion. Find the minimum sample size necessary.

liz/e -- please use the same name for your posts.

To find the minimum sample size necessary, we can use the formula for sample size calculation with proportions:

n = (Z^2 * p * (1-p)) / E^2

Where:
n is the minimum sample size needed.
Z is the Z-score corresponding to the desired confidence level. For a 90% confidence level, the Z-score is approximately 1.645.
p is the estimated proportion from previous studies (21% or 0.21).
E is the maximum allowable error (1% or 0.01).

Plugging in the values in the formula:

n = (1.645^2 * 0.21 * (1-0.21)) / 0.01^2

Simplifying:

n = (2.706025 * 0.21 * 0.79) / 0.0001

n = 0.4493656 / 0.0001

n ≈ 4493.656

Since the sample size must be a whole number, we need to round up to the nearest whole number to ensure we have enough participants. Therefore, the minimum sample size needed is 4494.

To find the minimum sample size necessary, we can use the formula for sample size estimation for a proportion, which is:

\[ n = \left(\frac{z^2 \cdot p \cdot (1-p)}{E^2}\right) \]

Where:
n = minimum sample size
z = z-value corresponding to the desired confidence level (90% confidence level corresponds to a z-value of 1.645)
p = estimated proportion of the population
E = maximum error tolerance (1% in this case, which is 0.01)

Given:
Confidence level = 90% (z = 1.645)
Estimated proportion (p) = 0.21 (21%)
Maximum error tolerance (E) = 0.01 (1%)

Plugging in the values into the formula:

\[ n = \left(\frac{1.645^2 \cdot 0.21 \cdot (1-0.21)}{0.01^2}\right) \]

\[ n = \left(\frac{1.645^2 \cdot 0.21 \cdot 0.79}{0.01^2}\right) \]

Simplifying the calculation:

\[ n = \left(\frac{1.645^2 \cdot 0.1659}{0.0001}\right) \]

\[ n \approx 913.28 \]

Rounding up to the nearest whole number, the minimum sample size necessary is 914. Therefore, the researcher would need to survey at least 914 young urban people (ages 21 to 35 years) to estimate the proportion of those who go to at least 3 concerts a year with a 90% confidence level and an accuracy within 1%.

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