a private opinion poll is conducted for a politician to determine why proportion of the population favors decriminalizing marijuana possession. how large a sample is needed in order to be 90 percent confident that the sample proportion will not differ from the true proportion by more than 2 percent
To determine the required sample size for a private opinion poll, you need to consider the confidence level, margin of error, and the population size.
1. Confidence level: In this case, it is specified as 90 percent confidence.
2. Margin of error: The question states that the sample proportion should not differ from the true proportion by more than 2 percent. This is the desired margin of error.
3. Population size: If the population size is very large (e.g., several millions), you can assume an infinite population size. However, if the population size is relatively small (e.g., a few thousand), you should adjust the calculation accordingly.
Given these details, here's how you can determine the required sample size:
Step 1: Find the Z-value corresponding to the desired confidence level. For a 90 percent confidence level, the Z-value is approximately 1.645. This value can be obtained from a standard normal distribution table.
Step 2: Calculate the square root of the product of the desired proportion (p) and the complementary proportion (1-p). Since the true proportion is unknown, it's common to assume a conservative estimate of 0.5 for p (since it will give the largest sample size).
Step 3: Determine the margin of error (E) by multiplying the Z-value from step 1 by the standard deviation from step 2.
Step 4: Use the formula for sample size:
Sample size (n) = [(Z * σ) / E]^2
Where:
- n represents the sample size
- Z is the Z-value
- σ is the standard deviation estimated in step 2
- E is the margin of error calculated in step 3
Note: If the population size (N) is relatively small, you should adjust the formula using a finite population correction factor:
Sample size (n) = [(Z * σ) / E]^2 * [(N - n) / (N - 1)]
Once all the values are substituted into the formula, you'll have the required sample size.
Remember, this calculation assumes a simple random sample, and it's important to carefully consider the representativeness of your sample to obtain accurate results.