Question

1a Scientists have speculated that eating a moderate amount of dark chocolate may increase the level of antioxidants, compounds that protect us against free radicals, which can cause heart disease and cancer. Levels of antioxidants in the bloodstream have been shown to increase significantly after consumption of dark chocolate. A study was designed, and the antioxidant concentration in patients who ate dark chocolate regularly was measured to test if there is evidence to suggest that the mean percentage of antioxidants in the bloodstream is greater than 0.4.

1. State the null and alternative hypotheses (2 pts):

2. Circle the correct type of hypotheses for this study, below (1 pt):

• An equal hypothesis versus a less-than hypothesis (left-tailed test)
• An equal hypothesis versus a greater-than hypothesis (right-tailed test)
• An equal hypothesis versus an unequal hypothesis (two-tailed test)

3. A random sample of antioxidant concentration in patients who ate dark chocolate provides enough evidence to conclude that the mean percentage of antioxidants in the bloodstream is greater than 0.4. Provide a conclusion statement for the study. (1 pt)

4. Suppose that the actual mean percentage of antioxidants in the bloodstream is 0.37. What type of error has been made (Type I or II) based on the conclusion in c.? Explain. (2 pts)

1b The annual Waikiki Roughwater Swim contest is held over a 2.4 mile course and ends near the Hilton Rainbow Tower. The 2008 winner was Trent Grimsey in 47 minutes, 59 seconds. Before the race, a random sample of the water current velocity (in knots) along the race course is obtained, and the resulting information is used to determine whether the race should be canceled. A mean current velocity, µ, of more than 0.65 knots is considered unsafe. The hypotheses are H₀: µ = 0.65 H₁ : µ > 0.65. If H₀ is rejected, the race will be canceled.

1. Suppose H₀ is true; the mean amount of the current velocity is 0.65. What are the consequences if H₀ is rejected? (1 pt)

2. Suppose H₀ is true; the mean amount of the current velocity is 0.65. What are the consequences if H₀ is not rejected? (1 pt)

3. Suppose H₁ is true; the mean amount of the current velocity is greater than 0.65. What are the consequences if H₀ is rejected? (1 pt)

4. Suppose H₁ is true; the mean amount of the current velocity is greater than 0.65. What are the consequences if H₀ is not rejected? (1 pt)

Answer 1

(first part)

(a.) State the null and alternative hypotheses (2 pts):

null hypothesis H0:=0.4

alternate hypothesis Ha: >0.4

since here we want to test the if there is evidence to suggest that the mean percentage of antioxidants in the bloodstream is greater than 0.4., so alternate hypothesis would be >0.4

(b.)Circle the correct type of hypotheses for this study, below (1 pt): right choice is second

An equal hypothesis versus a greater-than hypothesis (right-tailed test)

(c)A random sample of antioxidant concentration in patients who ate dark chocolate provides enough evidence to conclude that the mean percentage of antioxidants in the bloodstream is greater than 0.4. Provide a conclusion statement for the study. (1 pt) Since we reject the null hypothesis and conclude that eating a moderate amount of dark chocolate may increase the level of antioxidants, compounds that protect us against free radicals, which can cause heart disease and cancer.

(d) suppose that the actual mean percentage of antioxidants in the bloodstream is 0.37. What type of error has been made (Type I or II) based on the conclusion in c.? Explain. (2 pts)

Since here we reject the null hypothesis actually it was true. Here we committed Type I error

Type I error: Reject H0 when H0 is true

Type II error: Accept H0 when H0 is false

(Second part)

(a) Suppose H₀ is true; the mean amount of the current velocity is 0.65. What are the consequences if H₀ is rejected? (1 pt)

Type I error

(b)Suppose H₀ is true; the mean amount of the current velocity is 0.65. What are the consequences if H₀ is not rejected? (1 pt)

No error

(c) Suppose H₁ is true; the mean amount of the current velocity is greater than 0.65. What are the consequences if H₀ is rejected? (1 pt)

No error

(d) Suppose H₁ is true; the mean amount of the current velocity is greater than 0.65. What are the consequences if H₀ is not rejected? (1 pt)

Type II error