A national caterer determined that 87% of the people who sampled their food said that it was delicious. A random sample of 144 people is obtained from a population of 5000. The 144 people are asked to sample the caterer’s food. If p ^ is the sample proportion saying that the food is delicious, what is the mean of the sampling distribution of p ^ ?
mean is .87
The mean of the sampling distribution of p̂ can be calculated using the formula:
Mean (μ) = p,
where p is the population proportion.
Given that 87% of the people who sampled the food said it was delicious, the population proportion (p) is 0.87.
Therefore, the mean of the sampling distribution of p̂ is 0.87.
To find the mean of the sampling distribution, we need to use the formula for the mean of a proportion:
mean = p
In this case, p is the population proportion, which is 87% or 0.87. However, we only have a sample of 144 people from a population of 5000, so p ^ (pronounced "p-hat") is the sample proportion.
p ^ = x / n
where x is the number of people in the sample who said the food was delicious, and n is the sample size.
From the problem, we know that the sample size is 144. But we don't know the value of x.
To find the value of x, we need to know the number of people in the sample who said the food was delicious. Since we don't have this information, we can't calculate the exact mean of the sampling distribution.
However, if we assume that the sample is a simple random sample (meaning each person has an equal chance of being selected) and the sample size is large enough (greater than or equal to 30), we can make an approximation:
p ^ = p = 0.87
In this case, the mean of the sampling distribution would be:
mean ≈ p = 0.87
Therefore, the mean of the sampling distribution of p ^ (the sample proportion) is approximately 0.87.