Sampling Distribution of 𝒑̅

Question

Sampling Distribution of 𝒑̅

4. The population proportion is .45. What is the probability that a sample proportion will be within + .04 of the population proportion for each of the following sample sizes?
1. n=150
2. n=220
3. n=510
4. n=1200

Answer 1

To find the probability that a sample proportion will be within + .04 of the population proportion for each of the given sample sizes, we can use the sampling distribution of the sample proportion.

The sampling distribution of the sample proportion follows a normal distribution with a mean equal to the population proportion (p) and a standard deviation equal to the square root of (p * (1 - p) / n), where p is the population proportion and n is the sample size.

Let's calculate the probabilities for each sample size:

1. For n = 150:
The mean is p = 0.45.
The standard deviation is sqrt(0.45 * (1 - 0.45) / 150).

2. For n = 220:
The mean is p = 0.45.
The standard deviation is sqrt(0.45 * (1 - 0.45) / 220).

3. For n = 510:
The mean is p = 0.45.
The standard deviation is sqrt(0.45 * (1 - 0.45) / 510).

4. For n = 1200:
The mean is p = 0.45.
The standard deviation is sqrt(0.45 * (1 - 0.45) / 1200).

Now, we can use the Z-score formula to find the probability:

Z = (x - p) / σ

Where x is the value within + 0.04 of the population proportion, p is the population proportion, and σ is the standard deviation.

We will calculate the Z-score for x = p + 0.04 and x = p - 0.04, and use a standard normal table or calculator to find the corresponding probabilities.

Please provide the values of the population proportion (p) and sample size (n) for each question, and I can help you calculate the probabilities.

Answer 2

To find the probability that a sample proportion will be within + .04 of the population proportion, we need to make use of the sampling distribution of the sample proportion.

The sampling distribution of the sample proportion, denoted by 𝑝̂, follows a normal distribution if certain conditions are met: the sample is random, independent, and the sample size is large enough (n𝑝 ≥ 10 and n(1−𝑝) ≥ 10).

Once we determine that the conditions are met, we can calculate the mean and standard deviation of the sampling distribution.

The mean of the sampling distribution is equal to the population proportion 𝑝, and the standard deviation is given by the formula:
𝜎(𝑝̂) = sqrt(𝑝(1−𝑝)/𝑛)

To find the probability that a sample proportion will be within + .04 of the population proportion, we can use the standard normal distribution table or statistical software to calculate the probability.

Let's calculate the probability for each of the given sample sizes:

1. n = 150:
For this sample size, we can plug in the values 𝑝 = 0.45 and 𝑛 = 150 into the formula 𝜎(𝑝̂) = sqrt(𝑝(1−𝑝)/𝑛) to calculate the standard deviation of the sampling distribution.
Then, we can use the standard normal distribution table or statistical software to find the probability that 𝑝̂ will be within + .04 of the population proportion.

2. n = 220:
Follow the same steps as above but with 𝑛 = 220.

3. n = 510:
Again, follow the steps as above but with 𝑛 = 510.

4. n = 1200:
Repeat the steps as above but with 𝑛 = 1200.

By calculating the standard deviation and using the standard normal distribution table or statistical software, you can find the probability for each of the given sample sizes.