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Answer:
There is not a convincing statistical evidence, at the significance level of a = 0.05, that the manager's belief is correct.
As the null hypothesis maybe false accepting it makes a type II error.
Step-by-step explanation:
Let the null and alternate hypotheses be
H0: p ≤ 0.4 vs Ha : p>0.4
q= 1-p= 1-0.4= 0.6
The significance level alpha is 0.05
The critical region for one tailed test is Z> ± 1.645
The sample proportion is p^= x/n= 38/90=0.42222
Using the z statistic
z= p^- p/ √pq
z= 0.422-0.4/ √0.4*0.6
z= 0.04536
Since the calculated value of z= 0.04536 does not lie in the critical region Z> ± 1.645 we fail to reject null hypothesis.
There is not sufficient evidence to support the manager's claim.
Type I error is when we reject the true null hypothesis .
Type II error is when we accept the false null hypothesis .
As the null hypothesis maybe false accepting it makes a type II error.
There is not enough statistical evidence that the managers' claim is true.
What is the z test statistic for one sample proportion?
Suppose that we have:
- n = sample size
- [tex]\hat{p}[/tex] = sample proportion
- p = population proportion (hypothesised)
Then, the z test statistic for one sample proportion is:
[tex]Z = \dfrac{\hat{p} - p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
What are type I and type II errors?
The Type I error is false positive (returning the test result as positive, which means, rejecting the null hypothesis(conclusion of rejection of null hypothesis is called positive test result) when its incorrect).
The Type II error is false negative (returning the test result as negative, which means, accepting the null hypothesis(assuming negative test result, so no flaw in null hypothesis from the perspective of test) when its wrong).
For this case, we've got:
- Assumption by manager = Population proportion of repurchase by customers > 0.4
- Sample size = n = 90
- Sample proportion of repurchase = 38/90 ≈ 0.422
- Level of significance = α = 0.05
The hypotheses for test would be:
- Null hypothesis: Manager is not correct(nullifies managers' assumption): [tex]H_0: p \leq 0.4[/tex] (p is population proportion of repurchasing customers)
- Alternate hypothesis: Managers' assumption is true: [tex]H_1: p > 0.4[/tex]
It is single tailed.
At the level of significance 0.05, for single tail, the critical value of Z test statistic is found to be [tex]Z_{\alpha/2} = \pm1.645[/tex]
The z test statistic for one sample proportion is:
[tex]Z = \dfrac{\hat{p} - p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
We test null hypothesis, according to which p is 0.4 or less, so we get:
[tex]Z \approx \dfrac{0.422- 0.4}{\sqrt{\dfrac{0.4(1-0.4)}{90}}} \approx \dfrac{0.022}{\sqrt{0.00267}} \approx 0.426 < |Z_{\alpha/2}| = 1.645[/tex]
Thus, we accept the null hypothesis (we do so when [tex]Z < |Z_{\alpha/2}|[/tex] )
Therefore, there is not enough statistical evidence that the managers' claim is true.
Thus, here, we get:
- Type I error: Rejecting the fact that manager's claim is wrong(and thus accepting that manager's claim is right) when actually manager's claim is wrong.
- Type II error: Accepting the fact that manager's claim is wrong(and thus rejecting that manager's claim is right) when actually manager's claim is right.
Learn more about type I and type II errors here:
https://brainly.com/question/26067196
