Answer:
Step-by-step explanation:
Hello!
Your study variable is
X: Number of students that finished tha statistics course with an a, in a sample of 100 students.
This variable has a binomial distribution X~Bi(n;ρ)
The student believes that no more than 20% of the students pass the course with an A. This percentage is the population proportion symbolically: ρ ≤ 0.20, and is your null hypothesis., so:
a.
H₀: ρ ≤ 0.20
H₁: ρ > 0.20
α: 0,01
The statistic to use is the Z approximation for the proportions. To assemble this statistic, the central limit theorem is applied, this theorem allows us, at a sufficiently large sample size (n≥30), to approximate the distribution of the sample proportion (^p) to normal:
^p ≈ N(p; p(1-p)(1/n))
The statistic formula is:
Z=<u> ^p - p </u>≈ N(0;1)
√p(1-p)(1/n)
the sample proportion is ^p= 0.24
b.
Z=<u> 0.24 - 0.2 </u>= 1
√0.2*0.8(1/100)
The rejection region of this hypothesis is one-tailed (positive) If you ever have trouble identifying the type of rejection region look at the direction of the alternative hypothesis, if it has the symbol < then is a one-tailed, to the left, rejection region. If it has the symbol > then is a one-tailed, to the right, rejection region and if it has the ≠ symbol, it means the rejection region is "split" in two, i.e. two-tailed.
The critical value is:
If Z ≥ 2.33, then you reject the null hypothesis.
If Z < 2.33, then you do not reject the null hypothesis.
The decision is to not reject the null hypothesis.
c.
The p-value is defined as the probability corresponding to the calculated statistic if possible under the null hypothesis (i.e. the probability of obtaining a value as extreme as the value of the statistic under the null hypothesis).
Symbolically:
P(Z ≥ 1) = 1 - P(Z < 1) = 1 - 0.84134 = 0.15866
You have to look at what is the probability of the calculated Z value, the direction of the p-value is always the same as the rejection region. In this case, is a one-tailed p-value (to the right)
Using the p-value approach, the decision rule is always the same:
If p-value ≤ α, then you reject the null hypothesis.
If p-value > α, then you don't reject the null hypothesis.
Since the p-value 0.15866 > 0.01, then you do not reject the null hypothesis.
As expected, using the two methods you reached the same decision. If not then you have to check your maths.
I hope it helped!
