Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl). Assume that the population of x values has an approximately normal distribution.
9.3 9.0 10.3 8.9 9.4 9.8 10.0 9.9 11.2 12.1
(a) Use a calculator with mean and sample standard deviation keys to find the sample mean reading x and the sample standard deviation s. (Round your answers to two decimal places.)
x =______ mg/dl
s = _______ mg/dl
(b) Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood. (Round your answer to two decimal places.)
lower limit _______ mg/dl
upper limit _______ mg/dl
(a) To find the sample mean (x) and sample standard deviation (s), we can use a calculator with mean and sample standard deviation keys.
x = mean of the data set = sum of all the values / number of values
Using the calculator, we find:
x = 9.78 mg/dl (rounded to two decimal places)
s = sample standard deviation = square root of [(sum of (each value - mean)^2) / (number of values - 1)]
Using the calculator, we find:
s = 1.14 mg/dl (rounded to two decimal places)
Therefore, the sample mean reading (x) is 9.78 mg/dl, and the sample standard deviation (s) is 1.14 mg/dl.
(b) To find a 99.9% confidence interval for the population mean, we can use the formula:
Confidence interval = x ± (z * s / √n)
Where:
x = sample mean
z = z-score corresponding to the desired confidence level (in this case, 99.9%)
s = sample standard deviation
n = number of values
First, we need to find the z-score for a 99.9% confidence level. Using a z-score table or calculator, we find that the z-score for a 99.9% confidence is approximately 3.29.
Plugging in the values:
Confidence interval = 9.78 ± (3.29 * 1.14 / √10)
Calculating:
Confidence interval = 9.78 ± 3.69
Therefore, the 99.9% confidence interval for the population mean of total calcium is:
lower limit = 9.78 - 3.69 = 6.09 mg/dl (rounded to two decimal places)
upper limit = 9.78 + 3.69 = 13.47 mg/dl (rounded to two decimal places)
The lower limit is 6.09 mg/dl, and the upper limit is 13.47 mg/dl.
To solve this problem, we need to calculate the sample mean, sample standard deviation, and the confidence interval for the population mean. Here's how to do it:
(a) Calculating the sample mean (x) and sample standard deviation (s):
1. Add up all the calcium level readings:
9.3 + 9.0 + 10.3 + 8.9 + 9.4 + 9.8 + 10.0 + 9.9 + 11.2 + 12.1 = 99.9
2. Divide the sum by the number of readings (n) to find the sample mean (x):
x = 99.9 / 10 = 9.99 (rounded to two decimal places)
So, the sample mean is x = 9.99 mg/dl.
3. Subtract the sample mean from each reading, square the result, and sum up the squared differences:
(9.3 - 9.99)^2 + (9.0 - 9.99)^2 + (10.3 - 9.99)^2 + (8.9 - 9.99)^2 + (9.4 - 9.99)^2 + (9.8 - 9.99)^2 + (10.0 - 9.99)^2 + (9.9 - 9.99)^2 + (11.2 - 9.99)^2 + (12.1 - 9.99)^2 = 7.421
4. Divide the sum by (n-1) to find the sample variance (s^2):
s^2 = 7.421 / (10 - 1) = 0.8246 (rounded to four decimal places)
5. Take the square root of the sample variance to find the sample standard deviation (s):
s = √(0.8246) = 0.908 (rounded to three decimal places)
So, the sample standard deviation is s = 0.908 mg/dl.
Therefore, the sample mean is x = 9.99 mg/dl and the sample standard deviation is s = 0.908 mg/dl.
(b) Calculating the confidence interval:
To find a 99.9% confidence interval, we need to use the t-distribution with degrees of freedom (n-1). In this case, the degrees of freedom are 10 - 1 = 9.
Using a t-distribution table or a calculator, we find that the critical t-value for a two-tailed test at 99.9% confidence level with 9 degrees of freedom is approximately 3.250.
Next, we calculate the margin of error:
Margin of Error = (Critical t-value) * (Sample Standard Deviation / √n)
Margin of Error = (3.250) * (0.908 / √10) = 1.14 (rounded to two decimal places)
Finally, we calculate the confidence interval:
Lower Limit = Sample Mean - Margin of Error
Upper Limit = Sample Mean + Margin of Error
Lower Limit = 9.99 - 1.14 = 8.85 (rounded to two decimal places)
Upper Limit = 9.99 + 1.14 = 11.13 (rounded to two decimal places)
Therefore, the 99.9% confidence interval for the population mean of total calcium in this patient's blood is 8.85 mg/dl to 11.13 mg/dl.
Do not have calculator.
(a) Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
(b) Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.0005) and its Z score.
99.9% = mean ± Z SEm
SEm = SD/√n