I need help on this one too. Any one willing to get me started? THANKS!
Potassium is a mineral that helps the kidneys function normally. It also plays a key role in cardiac, skeletal, and smooth muscle contraction, making it an important nutrient for normal heart, digestive, and muscular function. A test for the level of potassium in the blood is not perfectly precise. Moreover, the actual level of potassium in a person’s blood varies slightly from day to day. Normal levels of potassium range from 3.7 mEq/L to 5.2 mEq/L.
(Note mEq/L represents milliequivalents per liter – which is just a type of concentration measurement.)
a) What does the phrase “statistically significant” mean? Explain.
b) If nine measurements for Jason were taken on different days with levels (mEq/L) given by 3.6, 3.7, 5.1, 5.6, 5.4, 4.7, 4.6, 3.8, and 5.2 respectively what is a 95% confidence level for his mean potassium level?
c) Do you think his potassium levels are normal? Explain.
This was already posted.
a) To understand what the phrase "statistically significant" means, we need to understand the concept of statistical significance in hypothesis testing. In statistical analysis, we often want to determine if there is a difference or relationship between two or more groups or variables. Statistical significance refers to the likelihood that any observed difference or relationship is not due to chance alone.
In other words, when we say a result is statistically significant, it means that the probability of obtaining such a result by chance alone, under the assumption that there is no true difference or relationship, is very low. Typically, a significance level, represented by alpha (α), is chosen beforehand (commonly 0.05 or 0.01). If the p-value, which measures the probability of observing the data given the null hypothesis, is less than the chosen significance level, then we say that the result is statistically significant, indicating that the observed difference or relationship is likely not due to chance alone.
b) To determine the 95% confidence interval for Jason's mean potassium level, we can calculate it using statistical methods. A confidence interval is a range of values that estimates the true population parameter (in this case, the mean potassium level) with a certain level of confidence. The standard formula for a confidence interval is:
Confidence interval = sample mean ± (critical value * standard error)
First, we calculate the sample mean of Jason's potassium levels by taking the average of the measurements: (3.6 + 3.7 + 5.1 + 5.6 + 5.4 + 4.7 + 4.6 + 3.8 + 5.2) / 9 = 4.8
Next, we need to determine the critical value, which depends on the desired confidence level. For a 95% confidence level, the critical value can be obtained from a t-distribution table. With 9 measurements, the degrees of freedom is 9-1 = 8. From the table, the critical value for a 95% confidence level with 8 degrees of freedom is approximately 2.306.
Now, we need the standard error, which measures the variability of the sample mean. The standard error can be calculated using the formula:
Standard error = standard deviation / √n
We do not have the standard deviation of the sample, so let's calculate it. The standard deviation is a measure of dispersion or spread around the mean. To calculate the sample standard deviation, we use the formula:
Standard deviation = √((Σ(xi - x̄)^2) / (n - 1))
Where xi represents each individual measurement, x̄ represents the sample mean, and n represents the number of measurements.
Using the given measurements, we find:
(3.6 - 4.8)^2 = 1.44
(3.7 - 4.8)^2 = 1.21
(5.1 - 4.8)^2 = 0.09
(5.6 - 4.8)^2 = 0.64
(5.4 - 4.8)^2 = 0.36
(4.7 - 4.8)^2 = 0.01
(4.6 - 4.8)^2 = 0.04
(3.8 - 4.8)^2 = 1.00
(5.2 - 4.8)^2 = 0.16
Summing these values, we get Σ(xi - x̄)^2 = 4.95
Now, using the formula for the standard deviation, we have:
Standard deviation = √(4.95 / (9 - 1)) ≈ 0.7416
Finally, calculating the standard error:
Standard error = 0.7416 / √9 ≈ 0.2472
Plugging the values into the confidence interval equation, we have:
Confidence interval = 4.8 ± (2.306 * 0.2472)
Confidence interval = 4.8 ± 0.5694
Confidence interval ≈ (4.23, 5.37)
Therefore, the 95% confidence level for Jason's mean potassium level is approximately between 4.23 mEq/L and 5.37 mEq/L.
c) To determine if Jason's potassium levels are normal, we compare his mean potassium level (4.8 mEq/L) to the normal range provided in the question (3.7 mEq/L to 5.2 mEq/L). From the 95% confidence level calculated in part b, we see that the interval (4.23, 5.37) falls entirely within the normal range.
Therefore, based on the given measurements and the calculated confidence interval, we can conclude that Jason's potassium levels are within the normal range.