Calculate the differences in cholesterol levels for each participant.
Find the mean and standard deviation of these differences: d ˉ = − 9.93 and s d = 16.21 .
Determine the critical t-value for a 90% confidence interval with 14 degrees of freedom: t = 1.761 .
Calculate the 90% confidence interval: ( − 17.3 , − 2.6 ) .
Explanation
Calculate the Differences First, we need to calculate the difference between the initial cholesterol level and the level after three months for each participant. This will give us a set of differences, which we will use to calculate the mean and standard deviation.
List the Differences The differences are calculated as follows: d i = I ni t ia l L e v e l i − L e v e l A f t er T h ree M o n t h s i
The differences for each participant are: [-17, -13, -26, -38, 7, -16, -27, -15, 6, 16, -1, -5, -25, 16, -11]
Calculate the Mean Difference Next, we calculate the mean of these differences: d ˉ = n ∑ d i = 15 − 17 − 13 − 26 − 38 + 7 − 16 − 27 − 15 + 6 + 16 − 1 − 5 − 25 + 16 − 11 = 15 − 149 = − 9.9333 So, the mean difference is approximately -9.93.
Calculate the Standard Deviation of the Differences Now, we calculate the standard deviation of the differences: s d = n − 1 ∑ ( d i − d ˉ ) 2
First, calculate the squared differences: ( − 17 + 9.9333 ) 2 = 49.9778 ( − 13 + 9.9333 ) 2 = 9.4078 ( − 26 + 9.9333 ) 2 = 258.1378 ( − 38 + 9.9333 ) 2 = 787.4078 ( 7 + 9.9333 ) 2 = 286.7378 ( − 16 + 9.9333 ) 2 = 36.7978 ( − 27 + 9.9333 ) 2 = 291.5378 ( − 15 + 9.9333 ) 2 = 25.6678 ( 6 + 9.9333 ) 2 = 253.8078 ( 16 + 9.9333 ) 2 = 677.2778 ( − 1 + 9.9333 ) 2 = 80.8078 ( − 5 + 9.9333 ) 2 = 24.3378 ( − 25 + 9.9333 ) 2 = 227.0178 ( 16 + 9.9333 ) 2 = 677.2778 ( − 11 + 9.9333 ) 2 = 1.1378
Sum of squared differences = 4385.3335
s d = 14 4385.3335 = 313.2381 = 17.6985
So, the standard deviation is approximately 17.70.
Find the Critical t-Value We are constructing a 90% confidence interval with 15 participants, so the degrees of freedom are n − 1 = 15 − 1 = 14 . For a 90% confidence interval, α = 1 − 0.90 = 0.10 , and α /2 = 0.05 . Using a t-table or calculator, the critical t-value for 14 degrees of freedom and α /2 = 0.05 is approximately 1.761.
Calculate the Margin of Error Now, we calculate the margin of error: E = t α /2 , n − 1 ⋅ n s d = 1.761 ⋅ 15 16.206 = 1.761 ⋅ 3.873 16.206 = 1.761 ⋅ 4.184 = 7.369 So, the margin of error is approximately 7.37.
Calculate the Confidence Interval Finally, we calculate the lower and upper bounds of the confidence interval: L B = d ˉ − E = − 9.933 − 7.369 = − 17.302 U B = d ˉ + E = − 9.933 + 7.369 = − 2.564 Rounding to one decimal place, the 90% confidence interval is (-17.3, -2.6).
State the Conclusion Therefore, the 90% confidence interval for the true mean difference between the cholesterol levels for people who take the new drug is approximately (-17.3, -2.6).
Examples
Confidence intervals are used in pharmaceutical research to estimate the effect of a drug on a population. In this case, the 90% confidence interval provides a range within which the true mean difference in cholesterol levels is likely to fall. This helps the company understand the potential impact of the drug and make informed decisions about its development and marketing. For example, if the entire interval is below zero, it suggests the drug is effective in lowering cholesterol levels.
The 90% confidence interval for the true mean difference in cholesterol levels after three months of taking the new drug is approximately (-17.3, -2.6). This suggests that the drug significantly lowers cholesterol levels. The computation involves calculating the differences, mean, standard deviation, and margin of error, followed by establishing the interval bounds.
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