Question

please solve all of the parts. thank you

2) A pharmaceutical company is formulating a new anti-obesity drug to target potential markets based on the population Body Mass Index (BMI) In a given country, official data claims that the average population BMI is about 25 kg/m2 with a standard deviation of 7 kg/m2. The company wishes to examine the extent up to which this claim is true and hence wants to test the Hypothesis Ho: p: 25 kg/m2 against lh: * 25 kg/m2 using the results of a random sample of 49 specimens. I- Find the type I error probability a if the acceptance area is defined as 23 s& s 27 IWhat is the probability of type II error B if the true BMI mean is 26 kg/m2 IlI- What is the probability of type II error B if the true BMI mean is 30 kg/m2 IV-Compare the two values of β calculated in the previous question, what conclusion can you draw?

Answer 1

Question 2

H₀ : $\mu$ = 25 kg/m²

H_a : $\mu \neq$ 25 kg/m²

standard deviation = $\sigma$ = 7 kg/m²

standard error of sample mean = $\sigma$/sqrt(n) = 7/sqrt(49) = 1 kg/m²

part -I

Here acceptance region is 23 $\leq \bar x \leq$ 27

Here type I error probability is when sample mean is out of the above given rejection region.

P( 23 $\leq \bar x \leq$ 27) = P( $\bar x \leq$ 27; $\mu$ = 25 ; se = 1) - P( $\bar x \leq$ 23; $\mu$ = 25 ; se = 1)

Z₂ = (27 - 25)/1 = 2

Z₁ = (23 - 25)/1 = -2

P( 23 $\leq \bar x \leq$ 27) = P(Z < 2) - P(Z < -2)

= 0.97725 - 0.02275 = 0.9545

Type I error = 1 - 0.9545 = 0.0455

(b) Now here true BMI = 26 kg/m²

so here type II error is when null hypothesis is false and we failed to reject the null hypothesis.

Tyep II error = P($\bar x \leq$ 27; 26 ; 1)

Z = (27 - 26)/1 = 1

Type II error = P(Z < 1) = 0.8413

(c)

Now here true BMI = 30 kg/m²

so here type II error is when null hypothesis is false and we failed to reject the null hypothesis.

Tyep II error = P($\bar x \leq$ 27; 30 ; 1)

Z = (27 - 30)/1 = -3

Type II error = P(Z < -3) = 0.00135

IV. Here there is a big difference in type II error for each part (ii) and part(iii). Here we draw the conclusion that if true BMI is too much higher than we can easily identify that we correctly failed to  reject the null hypothesis