Question

The city of Irvine reported that approximately 75% of residents are over the age of 60. Let X be the number of Irvine residents over the age of 60.From a random sample of 500 Irvine residents, 350 were over the age of 60.What is the sampling distribution of the sample proportion for the sample size of 500?Using the distribution of X from above, what is the probability that at most 350 of the 500 Irvine residents selected will be over the age of 60?What is the probability that at least 350 of the 500 residents in the sample were over the age of 60?What is the probability that between 400 and 475 of the residents were over the age of 60?

Answer 1

To find the sampling distribution of the sample proportion for a sample size of 500, you need to know the proportion of residents over the age of 60 in the population.

Step-by-step explanation:

To find the sampling distribution of the sample proportion for a sample size of 500, we need to know the proportion of residents over the age of 60 in the population. Since the city of Irvine reported that approximately 75% of residents are over the age of 60, we can assume that the true proportion is 0.75. The sampling distribution can be approximated by a normal distribution with mean equal to the true proportion, and standard deviation equal to sqrt(p*(1-p)/n), where p is the true proportion and n is the sample size.

Answer 2

The sampling distribution of the sample proportion can be approximated by a normal distribution, in this case with a mean of 0.75 and a standard deviation of 0.01633. To calculate probabilities, use the normal distribution function to find the area under the curve for the desired range of values.

Step-by-step explanation:

The sampling distribution for the sample proportion can be approximated by a normal distribution, since the sample size is large enough and the observations are independent. The mean of the sampling distribution of the sample proportion is equal to the population proportion, which is 0.75 in this case, and the standard deviation is given by:

Standard deviation = sqrt((p * (1-p)) / n)

where p is the population proportion and n is the sample size.

Using the given information, we can calculate the standard deviation:

Standard deviation = sqrt((0.75 * (1-0.75)) / 500) ≈ 0.01633

a) To find the probability that at most 350 of the 500 residents will be over the age of 60, we need to find the area under the sampling distribution curve from 0 to 350. This can be calculated using the normal distribution function.

b) To find the probability that at least 350 of the 500 residents will be over the age of 60, we need to find the area under the sampling distribution curve from 350 to 500. This can also be calculated using the normal distribution function.

c) To find the probability that between 400 and 475 of the residents will be over the age of 60, we need to find the area under the sampling distribution curve from 400 to 475. Again, this can be calculated using the normal distribution function.