Question

ABC Company has made a claim that, on average, their diet soda has less than one calorie. A consumer testing service examined nine cans, and the amount of calories recorded were 0.90, 0.95, 1.00, 1.05, 0.85, 1.00, 0.95, 0.95 and 0.90. Assume the underlying population is approximately normal. a. What are the sample mean and sample standard deviation, respectively?

b. Let μ denote the true mean calories of all diet sodas produced by ABC Company. What hypothesis should be tested in order to show if ABC’s claim is valid?

c. The appropriate statistic to test this hypothesis is ______

d. The value of the test statistic is ______

e. The appropriate number of degrees of freedom for this test statistic is ______

f. What conclusion can be made with 95% confidence?

Answer 1

Answer:

a. Sample mean is 0.95  and Sample standard deviation is ≈0.058

b. $H_{0}$: μ=1 calorie  and $H_{a}$: μ<1 calori

c. The appropriate statistic to test this hypothesis is t-statistic.

d. The value of the test statistic is -2,586

e. The appropriate number of degrees of freedom for this test statistic is 8

f. In 95% confidence, the claim that, on average, their diet soda has less than one calorie is true.

Step-by-step explanation:

a.

Sample mean is 0.95 (0.90+ 0.95+1.00+ 1.05+ 0.85+1.00+ 0.95+0.95+0.90)÷9

Sample standard deviation is ≈0.058 (sum of square differences from the mean)

b.

μ denote the true mean calories of all diet sodas produced by ABC Company.

Then

$H_{0}$: μ=1 calorie

$H_{a}$: μ<1 calori

c.

The appropriate statistic to test this hypothesis is t-statistic.

d.

The value of the test statistic is

t=$\frac{X-M}{\frac{s}{\sqrt{N} } }$ where

• X = sample mean (0.95)

• M is the mean amount of calorie assumed under null hypothesis (1)

• s is the sample standard deviation (0.058)

• N is the sample size (9)

t=$\frac{0.95-1}{\frac{0.058}{\sqrt{9} } }$ ≈ -2,586

e.

The appropriate number of degrees of freedom for this test statistic is 8

f.

in 95% confidence, the result is significant, since one tail t(critical) with 8 degrees of freedom is bigger than t(sample) i.e. -1.860 > -2,586

We reject the null hypothesis in favor of the alternative hypothesis. Therefore the claim that, on average, their diet soda has less than one calorie is true.