Determine the sample size required to estimate a population proportion to within 0.032 with 95.7% confidence, assuming that you have no knowledge of the approximate value of the sample proportion.

Sample Size =

B. Repeat part the previous problem, but now with the knowledge that the population proportion is approximately 0.33.

Sample Size =

i don't understand how you can do this without the margin or error.

Try this formula:

n = [(z-value)^2 * p * q]/E^2

Note: n = sample size needed; .5 for p and .5 for q are used if no value is stated in the problem. E = maximum error, which is 0.032. Z-value is found using a z-table (for 95.7% confidence).

For B, use 0.33 for p and 0.67 for q.
Note: q = 1 - p

I'll let you take it from here.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 11 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with σ = 0.30 gram.

Find the sample size necessary for an 80% confidence level with a maximal error of estimate E = 0.09 for the mean weights of the hummingbirds. (Round up to the nearest whole number.)

To determine the sample size required to estimate a population proportion with a given margin of error and confidence level, you can use the formula:

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

Where:
- Z is the z-score corresponding to the desired confidence level
- p is the estimated population proportion
- E is the desired margin of error

In the first problem, since you have no knowledge of the approximate value of the sample proportion, you can assume the most conservative value of p = 0.5. The z-score corresponding to a 95.7% confidence level is approximately 1.96.

So, for the first problem, the sample size can be calculated as:

Sample size = (1.96^2 * 0.5 * (1 - 0.5)) / (0.032^2)
Sample size = 9604.95

Therefore, the sample size required to estimate the population proportion to within 0.032 with 95.7% confidence, assuming no knowledge of the approximate value of the sample proportion, is approximately 9605.

In the second problem, since you have knowledge that the population proportion is approximately 0.33, you can use this value for p.

Using the same formula, the sample size can be calculated as:

Sample size = (1.96^2 * 0.33 * (1 - 0.33)) / (0.032^2)
Sample size = 925.25

Therefore, the sample size required to estimate the population proportion to within 0.032 with 95.7% confidence, assuming the population proportion is approximately 0.33, is approximately 926.

I hope this clarifies the process for you! Let me know if you have any more questions.

To determine the sample size required to estimate a population proportion with a given level of confidence, you can use the formula:

Sample size = (Z^2 * p * q) / E^2,

where:
- Z is the Z-score corresponding to the desired level of confidence,
- p is the estimated population proportion,
- q is the complementary probability of p (1-p), and
- E is the maximum desired margin of error.

In the first problem, it is stated that you have no knowledge of the approximate value of the sample proportion. Therefore, you need to assume the most conservative estimate for p, which is 0.5. This assumption ensures that the sample size obtained will be large enough regardless of the actual value of p.

Given:
- Level of confidence: 95.7% (which corresponds to a Z-score of approximately 1.96)
- Maximum margin of error: 0.032
- p (estimated population proportion) = 0.5

Using the formula, let's calculate the sample size:
Sample size = (1.96^2 * 0.5 * (1-0.5)) / 0.032^2

Once you plug in the values into the formula and calculate, you will get the sample size required to estimate the population proportion within the desired margin of error, with the given level of confidence.

In the second problem, you now have knowledge of the approximate value of the population proportion, which is given as 0.33.

Given:
- Level of confidence: 95.7% (which corresponds to a Z-score of approximately 1.96)
- Maximum margin of error: 0.032
- p (estimated population proportion) = 0.33

Using the same formula as before, but now with the known value of p, you can calculate the sample size:
Sample size = (1.96^2 * 0.33 * (1-0.33)) / 0.032^2

By plugging in the values and calculating, you will obtain the required sample size to estimate the population proportion within the desired margin of error, assuming the known approximate value of the population proportion.