QuestionQuestion 5, 10.1.13
Part 1 of 5
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For students who first enrolled in two-year public institutions in a recent semester, the proportion who earned a bachelor's degree within six years was 0.393 . The president of a certain junior college believes that the proportion of students who enroll in her institution have a lower completion rate.
(a) State the null and alternative hypotheses in words.
(b) State the null and alternative hypotheses symbolically.
(c) Explain what it would mean to make a Type I error.
(d) Explain what it would mean to make a Type II error.
(a) State the null hypothesis in words. Choose the correct answer below.
A. Among students who enroll at the certain junior college, the completion rate is less than 0.393.
B. Among students who enroll at the certain junior college, the completion rate is greater than 0.393.
C. Among students who enroll at the certain junior college, the completion rate is 0.393 .
D. Among students who first enroll in two-year public institutions, the completion rate is 0.393 .
Studdy Solution
STEP 1
What is this asking?
We want to figure out how to set up the *hypothesis test* to see if the junior college's completion rate is actually lower than the national average.
Watch out!
Don't mix up the *null hypothesis* and the *alternative hypothesis*!
Also, be super careful about what a *Type I error* and a *Type II error* actually mean.
STEP 2
1. State the hypotheses in words.
2. State the hypotheses symbolically.
3. Explain Type I error.
4. Explain Type II error.
STEP 3
The *null hypothesis* is like our starting assumption.
It's what we assume is true unless we have enough evidence to say otherwise.
In this case, the null hypothesis is that the junior college's completion rate is the **same** as the national average, which is **0.393**.
So, in words: *Among students who enroll at the certain junior college, the completion rate is 0.393*.
STEP 4
The *alternative hypothesis* is what we're trying to find evidence for.
Here, we want to see if the junior college has a *lower* completion rate than the national average.
So, in words: *Among students who enroll at the certain junior college, the completion rate is less than 0.393*.
STEP 5
Let represent the **true proportion** of students at the junior college who earn a bachelor's degree within six years.
STEP 6
Symbolically, the null hypothesis is written as .
This means we're *assuming* the junior college's completion rate is **equal** to the national average.
STEP 7
Symbolically, the alternative hypothesis is written as .
This is what we're *testing* – whether the junior college's completion rate is **less than** the national average.
STEP 8
A *Type I error* happens when we **reject** the null hypothesis when it's actually **true**.
It's like saying the junior college has a lower completion rate when, in reality, it's the same as the national average.
STEP 9
In this case, a Type I error would mean concluding that the junior college has a lower completion rate than the national average of **0.393**, when in fact, its true completion rate is **equal** to **0.393**.
STEP 10
A *Type II error* happens when we **fail to reject** the null hypothesis when it's actually **false**.
It's like saying the junior college's completion rate is the same as the national average when, in reality, it's lower.
STEP 11
In this case, a Type II error would mean concluding that the junior college's completion rate is **not lower** than the national average of **0.393**, when in fact, its true completion rate is actually **less than 0.393**.
STEP 12
(a) C.
Among students who enroll at the certain junior college, the completion rate is 0.393.
(b) ,
(c) Concluding the junior college has a lower completion rate when it's actually the same as the national average.
(d) Concluding the junior college's completion rate is not lower than the national average when it actually is lower.
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