A random sample of the houses in a particular city is selected and the level of radon gas is measured for each house in the sample. The values collected are given below in parts per million (ppm). Experience has shown that radon gas level is approximately normally distributed for this population.

4, 8, 13, 6, 7, 3, 9, 8, 11, 8

a) Calculate a point estimate of the population mean of radon gas levels. (5 pts)

b) Construct and interpret a 99% confidence interval for . (6 pts)

c) The city standards for radon call for no more than a level of 6 ppm. From these data, does it appear that the mean radon level for city houses is under 6 ppm? Provide statistical justification for your response. (2 pts)

a) To calculate a point estimate of the population mean of radon gas levels, we need to calculate the sample mean. The point estimate of the population mean is equal to the sample mean.

To calculate the sample mean, we add up all the values and divide by the number of values.

Sum of values = 4 + 8 + 13 + 6 + 7 + 3 + 9 + 8 + 11 + 8 = 77
Number of values = 10

Sample Mean = Sum of values / Number of values = 77 / 10 = 7.7

Therefore, the point estimate of the population mean of radon gas levels is 7.7 ppm.

b) To construct a confidence interval for the population mean, we need to determine the margin of error and then calculate the lower and upper bounds.

The formula for the margin of error is:

Margin of Error = Critical value * Standard deviation of the sample mean

Since the population standard deviation is not given, we need to estimate it using the sample standard deviation. The formula for the sample standard deviation is:

Sample standard deviation = sqrt((sum of (value - sample mean)^2) / (number of values - 1))

Using the given values, we can calculate the sample standard deviation:

Sample standard deviation = sqrt(( (4-7.7)^2 + (8-7.7)^2 + (13-7.7)^2 + (6-7.7)^2 + (7-7.7)^2 + (3-7.7)^2 + (9-7.7)^2 + (8-7.7)^2 + (11-7.7)^2 + (8-7.7)^2 ) / (10-1))

= sqrt((36.09 + 0.09 + 20.49 + 2.89 + 0.49 + 23.29 + 1.69 + 0.09 + 11.56 + 0.09) / 9)

= sqrt(97.77 / 9)

= sqrt(10.8642...)

≈ 3.295

Now, we need to determine the critical value. The critical value is based on the desired confidence level and the sample size. Since the confidence level is 99% and the sample size is 10, we need to find the critical value from the t-distribution table.

Using the t-distribution table with 9 degrees of freedom (n-1), we find the critical value to be approximately 3.250.

Plugging in the values:

Margin of Error = Critical value * Standard deviation of the sample mean

= 3.250 * (3.295 / sqrt(10))

≈ 3.250 * (3.295 / 3.162)

≈ 3.250 * 1.042

≈ 3.384

The margin of error is approximately 3.384.

To calculate the lower and upper bounds of the confidence interval, we subtract and add the margin of error to the sample mean:

Lower bound = Sample mean - Margin of error = 7.7 - 3.384 = 4.316

Upper bound = Sample mean + Margin of error = 7.7 + 3.384 = 11.084

Therefore, the 99% confidence interval for the population mean is approximately (4.316, 11.084).

c) The city standards for radon call for no more than a level of 6 ppm.

From the confidence interval obtained in part b, we can see that the lower bound of the interval is 4.316. Since this value is less than 6, it suggests that it is possible that the mean radon level for city houses is under 6 ppm.

However, to provide statistical justification, we need to check if 6 ppm falls within the confidence interval.

Since 6 falls within the range of the confidence interval (4.316, 11.084), it suggests that the mean radon level for city houses could potentially be 6 ppm or lower.

Therefore, there is statistical justification to conclude that the mean radon level for city houses is likely under 6 ppm based on the given data and confidence interval.