Question

6. (30 points) It was traditionally accepted that you could get 120 calories on average from a serving of vanilla yogurt. In 2016 a new diet guide claims that you will get more than calories on average. Is there any compelling evidence to support this clairm data provided in problem 1 to conduct a test, with a 0.05. Assume the population is normally distributed. (a) (5 points) Please write appropriate hypotheses. (b) (5 points) Compute the test statistic. (c) (5 points) Construct rejection region and make the decision. (d) (5 points) Explain what are type 1 and type Ⅱ errors under the context. (e) (5 points) What is the probability of making type I error? (f) (5 points) What type of error could we possibly make? Type I error or type II error?

Answer 1

first of all you have not provided the data regarding this problem and you haven't specified the sample size for the experiment. So i can give a raw overview of the problem.

a) $H0:\theta =120; H1:\theta >120$

where theta is the mean calorie

(b) test statistics:

1)if sample size is <30 and samples are drawn from a normal population and SD is unknown, you can use t-test for single mean

$Z=xbar-\mu /(S/\sqrt{n})$

S= sample SD

mu=120

xbar,S= calculate from sample

2) if sample size is >30 and samples itself are normal and population SD is known, you can use  large sample normal test for single mean

$Z=xbar-\mu /(\sigma /\sqrt{n})$

sigma= population SD

xbar=calculate from sample

mu=120

c) IF value calculated from test statistics (calculated value) exceeds the tabulated value(from normal/t distribution table) then

REJECT H0

d)

typel- P(reject HOwhen HOistrue)-a

It the prob. of WRONG REJECTION

$type2=P(accept H0 when H0 is false)=\beta$

It the prob. of WRONG ACCEPTANCE