Question

Some dietitians have suggested that highly acidic diets can have an adverse affect on bone density in humans. Alkaline diets have been marketed to avoid or counteract this effect. Veterinary researchers wondered if the same thing true for cats, and whether an alkaline diet might be beneficial. Two groups of 4 cats each were fed diets for 12 months that differed only in acidifying or alkalinizing properties. The bone mineral density (g/cm2) of each cat was measured at the end of 12 months. It is known from many previous studies that bone mineral density in cats varies with σ = 0.04 g/cm2. The researchers found that the 4 cats on the acidifying diet had a mean bone mineral density of 0.68 g/cm2, while the 4 cats on the alkalinizing diet had a mean bone mineral density of 0.71 g/cm2. Does this show a significant difference in mean bone mineral density between the two diets?

The solution must include the following parts, in order. Point form is fine.

a) First answer each of these questions:

i) Does this call for a confidence interval or a hypothesis test?

ii) Is this 1 sample or 2 samples?

iii) Is this about mean(s) or proportion(s)?

iv) If this is about mean(s), do you know the SD of the population (σ — or σ1 & σ2 )? (If this is about proportions, skip this question.)

b) What are the population parameter(s), and what are the sample statistic(s)? Provide symbols, and say what they represent in the context of the question. (For example, "µ is the mean completion time of the population, and x̄ is the mean completion time for the sample".) You may want to give the statistic values here.

c) Complete the question. Details depend on the kind of problem:

• For a confidence interval:

i) State the confidence level. (If it is not given, make a reasonable choice.)

ii) Give the formula for the margin of error (symbols only, no numbers!).

iii) Calculate the margin of error (show your work!).

iv) State the confidence interval in a complete sentence (in words!), in the context of the original problem. (You may use whichever form you prefer.)

• For a hypothesis test:

i) State the significance level (alpha). (If it is not given, make a reasonable choice.)

ii) Give the formula for the test statistic (z or t) (symbols only, no numbers!).

iii) State the null and alternative hypotheses. Use symbols, and state them in the context of the original problem. (A sketch is optional, but very useful.)

iv) Calculate the test statistic (z or t) (show your work!), and determine the p-value.

v) State your conclusion in a complete sentence (in words!), in the context of the original problem. Your conclusion should state whether or not you reject the null hypothesis, and what this says about the original question.

Answer 1

(a)

(i) Hypothesis test

(ii) Two samples

(iii) Means

(iv) Yes, σ1 = σ2 = 0.04

(b)

n1 = 4, n2 = 4, x1-bar = 0.68, x2-bar = 0.71, σ1= 0.04, σ2 = 0.04, μd = Mean difference to be tested = μ1 - μ2 = 0

(c)

(i) α = 0.05 (assumed)

(ii) z = (x1-bar -x2-bar)/√{(σ1^2 /n1) + (σ2^2 /n2)}

(iii) Ho: μ1 - μ2 = 0 and Ha: μ1 - μ2 ≠ 0

(iv) z = (0.68 - 0.71)/√(((0.04)^2/4) + ((0.04)^2/4) = -1.0607

p- value = 0.288844366   

(v) Since 0.2888 > 0.05, we fail to reject Ho. There is no sufficient evidence of a significant difference in mean bone mineral density between the two diets.

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