Question

Round to the fourth and write as a proportion unless otherwise stated.

Problem 2

A food chemist analyzed the calorie content for a popular type of chocolate cookie. Following are the
number of calories in a random sample of eight cookies. Calorie content follows a Normal Distribution.

113 114 111 116 115 120 118 116

a. For a 95% confidence interval, find the margin of error for the mean number of calories in this type of cookie. Show your work and calculations.
b. Build and interpret a 95% confidence interval.
c. Another chemist claims that the cookie has an average calorie content of exactly 114 calories. Respond to the chemist using your interval from part b. Make your response understandable to someone not in the class.
d. Respond to the statement that the average calorie content is less than 120 calories. Use your interval in part b and make your response understandable to someone not in the class.

Answer 1

Solution

Let X = calorie content for a popular type of chocolate cookie

Given X ~ N(μ, σ²),

Back-up Theory

100(1 - α) % Confidence Interval for the mean μ, when σ is not known is: Xbar ± MoE    ………. (1)

where

MoE = (t_{n- 1, α /2})s/√n ………………………………………………………………………………. (2)

with

Xbar = sample mean,

t_{n – 1, α /2} = upper (α/2)% point of t-distribution with (n - 1) degrees of freedom,

s = sample standard deviation and

n = sample size.

Application of confidence interval to take decision on the hypothesis

For any population parameter θ, Null hypothesis: H₀: θ = θ₀ Vs Alternative: H₁: θ ≠ θ₀ is rejected at significance level, α%, if θ₀ is not contained in the 100(1 – α)% Confidence Interval for θ. …… (3)

Now to work out the solution,

Final answers are first given. Detailed working follows at the end.

Part (a)

Vide (2), margin of error for the mean number of calories in this type of cookie is:

18.0994 Answer

Part (b)

95% confidence interval is: [104.0994, 140.9743]   Answer

Part (c)

Vide (3), since the above confidence interval holds 114, claim of another chemist that the cookie has an average calorie content of exactly 114 calories is valid. Answer

Part (d)

With respect to the above confidence interval, 120 is 16 calories more than the lower bound, but 20 calories less than the upper bound. This implies that the actual mean is more likely to be above 120 than less than 120. Answer

Details of Calculations

n	8
Xbar	123
s	21.6494
√n	2.828427
α	0.05
n - 1	7
tα/2	2.364624
(s/√n)(tα/2)	18.09935
Lower Bound	104.7757
Upper Bound	140.9743

DONE