A random sample of the houses in a particular city is selected and each of the houses in a particular city is selected and each of the house's level of radon gas is measured. The values collected are given in the table below in parts per million (ppm).
4 8 13 6 7 3 9 8 11 8
a. Give a point estimate for u=the average level of radon gas in a house in this particular city.
b. Let pi=the proportion of houses in this city having radon levels exceeding 8 ppm. Give a point estimate of pi.
c. Give a point estimate of the median level or radon gas in a house in this city.
d. Give a point estimate of the standard deviation of the level of radon gas in a house in this city.
e. What conditions are required in order to have a valid confidence interval for u in this data set?
aa
a. To give a point estimate for the average level of radon gas in a house in this particular city, you would need to calculate the sample mean. To do this, add up all the values and divide the sum by the number of values in the sample.
For the given data set:
4 + 8 + 13 + 6 + 7 + 3 + 9 + 8 + 11 + 8 = 77
There are 10 values in the sample, so the sample mean is:
77 / 10 = 7.7 ppm
Therefore, the point estimate for the average level of radon gas in a house in this particular city is 7.7 ppm.
b. To give a point estimate of pi, the proportion of houses in this city having radon levels exceeding 8 ppm, you would need to calculate the proportion of values in the sample that exceed 8 ppm. Count the number of values above 8 ppm and divide it by the total number of values in the sample.
In the given data set, there are 4 values (8, 13, 9, 11) that exceed 8 ppm. Therefore, the proportion estimate of pi is:
4 / 10 = 0.4 or 40%
Hence, the point estimate of the proportion of houses in this city having radon levels exceeding 8 ppm is 40%.
c. To give a point estimate of the median level of radon gas in a house in this city, you would need to arrange the values in ascending order and find the middle value. In this case, since there are 10 values, the median will be the average of the fifth and sixth values.
First, let's sort the values in ascending order: 3, 4, 6, 7, 8, 8, 8, 9, 11, 13
The fifth value is 8, and the sixth value is 8 as well. Therefore, the median point estimate is:
(8 + 8) / 2 = 8 ppm
Therefore, the point estimate of the median level of radon gas in a house in this city is 8 ppm.
d. To give a point estimate of the standard deviation of the level of radon gas in a house in this city, we need to calculate the sample standard deviation. The formula for sample standard deviation is a bit complex, but here are the steps:
1. Calculate the mean (which we already found to be 7.7).
2. Subtract the mean from each value and square the result.
3. Sum up all the squared differences.
4. Divide the sum by n-1 (where n is the number of values in the sample).
5. Take the square root of the division result.
Using the given data set, we can perform these calculations:
Step 1: Find the mean (which is 7.7).
Step 2: Subtract mean and square the differences for each value:
(4-7.7)^2 = 13.69
(8-7.7)^2 = 0.09
(13-7.7)^2 = 34.81
(6-7.7)^2 = 2.89
(7-7.7)^2 = 0.49
(3-7.7)^2 = 19.36
(9-7.7)^2 = 1.69
(8-7.7)^2 = 0.09
(11-7.7)^2 = 10.89
(8-7.7)^2 = 0.09
Step 3: Sum up all the squared differences:
13.69 + 0.09 + 34.81 + 2.89 + 0.49 + 19.36 + 1.69 + 0.09 + 10.89 + 0.09 = 83.10
Step 4: Divide the sum by n-1:
83.10 / (10-1) = 9.23
Step 5: Take the square root of the division result:
√9.23 ≈ 3.04
Therefore, the point estimate of the standard deviation of the level of radon gas in a house in this city is approximately 3.04 ppm.
e. In order to have a valid confidence interval for u (the average level of radon gas), several conditions need to be met. These conditions are:
1. Random Sampling: The sample needs to be selected randomly from the population to ensure that it is representative of the entire population of houses in the city.
2. Normality of Data or Sufficient Sample Size: The data should follow an approximately normal distribution, or if the sample size is large enough (typically n ≥ 30), the Central Limit Theorem can be applied, allowing for a valid confidence interval even if the data is not normally distributed.
3. Independence: The measurements should be independent of each other, meaning that the level of radon gas in one house should not influence the level of radon gas in another house.
If these conditions are met, you can proceed to calculate a confidence interval for u using appropriate statistical methods.