Exercises 20.46-20.58 require the use of a computer and...

Exercises $20.46-20.58$ require the use of a computer and software. Use a $5 \%$ significance level unless specified otherwise. High levels of cholesterol can lead to heart attacks. For those with very high levels (above $7.5 \mathrm{mmol} / \mathrm{L}$ ), doctors prescribe drugs to reduce cholesterol levels. A pharmaceutical company has recently developed four such drugs. An experiment was conducted to determine whether any differences exist in their benefits. The company selected 25 groups of four men, each of whom had cholesterol levels in excess of $7.5 \mathrm{mmol} / \mathrm{L}$. In each group, the men were matched according to age and weight. The drugs were administered over a 2-month period, and the reduction in cholesterol was recorded. Do these results allow the company to conclude that differences exist between the four new drugs?

Exercises $20.46-20.58$ require the use of a computer and software. Use a $5 \%$ significance level unless specified otherwise.
High levels of cholesterol can lead to heart attacks. For those with very high levels (above $7.5 \mathrm{mmol} / \mathrm{L}$ ), doctors prescribe drugs to reduce cholesterol levels. A pharmaceutical company has recently developed four such drugs. An experiment was conducted to determine whether any differences exist in their benefits. The company selected 25 groups of four men, each of whom had cholesterol levels in excess of $7.5 \mathrm{mmol} / \mathrm{L}$. In each group, the men were matched according to age and weight. The drugs were administered over a 2-month period, and the reduction in cholesterol was recorded. Do these results allow the company to conclude that differences exist between the four new drugs?

Key Concepts

Analysis of Variance (ANOVA)

Analysis of variance (ANOVA) is a statistical technique used to compare the means of three or more groups to ascertain if at least one of the group means is statistically different from the others. It partitions the overall variability into variability between groups and within groups, and its F-test helps determine if the treatment effects are significant.

Randomized Block Design

A randomized block design is an experimental design that seeks to reduce variability by grouping subjects into blocks that are similar based on characteristics such as age and weight. Each block then receives all treatments, which allows for more precise comparisons by accounting for potential confounding variables within the groups.

Hypothesis Testing

Hypothesis testing is a statistical method used to infer whether there is sufficient evidence from sample data to conclude that a certain condition is true for the entire population. In the context of comparing multiple treatments or drugs, the null hypothesis typically asserts that all treatments have the same effect, while the alternative hypothesis posits that at least one treatment differs in its impact.

Significance Level

The significance level is the threshold used to determine whether to reject the null hypothesis, commonly denoted as alpha (?). A 5% significance level means that there is a 5% risk of committing a Type I error, which involves concluding that a difference exists when in fact no true difference is present.

Transcript

00:01 At two weeks after january 1, 2016, a patient's total cholesterol level t of t and good cholesterol level g of t are measured in milligrams per deciliter of blood.

00:12 The cholesterol ratio r of t equals g of t over t of t is used to gauge the safety of a patient's cholesterol risk.

00:21 The risk of cholesterol -related illness is minimized when r is greater than one -fifth.

00:27 A.

00:28 How is it possible for a patient's total cholesterol risk? to increase while the cholesterol ratio remains constant.

00:38 If the total cholesterol of the patient is increasing, then we'll know that t prime of t is greater than 0.

00:48 If the cholesterol ratio is constant, we'll know that r prime of t is equal to 0.

00:57 Let's see what's happening with g prime of t.

01:02 To do that, we'll need to find the derivative r prime of t because this r t equals g of t over t of t is a quotient we'll need to use the quotient rule.

01:15 The quotient rule tells us that to take the derivative of a ratio of functions y over z, we will take the derivative of the numerator multiplied by the denominator minus the function in the numerator multiplied by the derivative of the function in the denominator all over the denominator squared.

01:41 So when we apply this format to our equation for r of t, you will have r prime of t is equal to g prime of t times t of t minus g of t times t times t prime of t all over t of t squared.

02:11 Now we know that r prime of t is equal to zero, so let's just set this equal to zero.

02:17 If we isolate this g prime of t, then we'll find that g prime of t is equal to g of t times t prime of t over t of t.

02:37 We know that r of t is equal to of t, so we could also rewrite this as r of t times t prime of t.

02:49 So, it's possible for a patient's total cholesterol to increase while the cholesterol ratio remains constant if the good cholesterol level changes at a rate equal to r of t t times t prime of t.

03:14 B.

03:15 On january 1st, 2016, the patient's total cholesterol is at 120 milligrams per deciliter, and their good cholesterol is at 30 milligrams per deciliter.

03:26 R is greater than 1 5th, but the doctor wants the patient's good cholesterol to increase to 40 milligrams per deciliter, so the patient starts a diet which will increase their good cholesterol by 1 milligram per deciliter each week, but will not change the cholesterol ratio.

03:43 During the first week of the diet, what is the rate of change of the patient's total cholesterol? let's pull out all of our relevant information.

03:51 So on january 1st, 2016.

03:55 This is 0 weeks after january 1st, 2016, so we know that t equals 0...

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