Math / Data & Statistics

QuestionThe use of the formula for margin of error requires a large sample. For each of the following combinations of $n$ and $\hat{p}$ , indicate whether the sample size is large enough for use of this formula to be appropriate. (a) $n=100$ and $\hat{p}=0.60$ The sample size is large enough. The sample size is not large enough. (b) $n=40$ and $\hat{p}=0.25$ The sample size is large enough. The sample size is not large enough. (c) $n=50$ and $\hat{p}=0.25$ The sample size is large enough. The sample size is not large enough. (d) $n=80$ and $\hat{p}=0.10$ The sample size is large enough. The sample size is not large enough.

Studdy Solution

STEP 1

What is this asking? Do we have enough data to use a specific formula about margin of error? Watch out! Don't forget, a large enough sample size depends on both $n$ and $\hat{p}$ !

STEP 2

1. Check Sample Size

STEP 3

Alright, let's dive into this!
To figure out if our sample size is large enough, we need to check if both $n\hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 10.
This magic number 10 ensures we have enough data to make reliable estimates.

STEP 4

Let's calculate $n\hat{p}$ : $100 \cdot 0.60 = 60$ .
This is greater than 10, awesome!
Now, let's check $n(1-\hat{p})$ : $100 \cdot (1-0.60) = 100 \cdot 0.40 = 40$ .
This is also greater than 10, fantastic!

STEP 5

Time for $n\hat{p}$ : $40 \cdot 0.25 = 10$ .
Right on the edge, but it works!
Now for $n(1-\hat{p})$ : $40 \cdot (1-0.25) = 40 \cdot 0.75 = 30$ .
Greater than 10, we're good to go!

STEP 6

Let's compute $n\hat{p}$ : $50 \cdot 0.25 = 12.5$ .
Greater than 10, looking good!
Now for $n(1-\hat{p})$ : $50 \cdot (1-0.25) = 50 \cdot 0.75 = 37.5$ .
Also greater than 10, perfect!

STEP 7

Calculating $n\hat{p}$ : $80 \cdot 0.10 = 8$ .
Uh oh, this is less than 10.
We don't even need to check the second condition, since this one already fails.

STEP 8

(a) The sample size is large enough. (b) The sample size is large enough. (c) The sample size is large enough. (d) The sample size is not large enough.

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Studdy Solution

STEP 1

What is this asking? Do we have enough data to use a specific formula about margin of error? Watch out! Don't forget, a large enough sample size depends on both $n$ and $\hat{p}$ !

STEP 2

1. Check Sample Size

STEP 3

Alright, let's dive into this!
To figure out if our sample size is large enough, we need to check if both $n\hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 10.
This magic number 10 ensures we have enough data to make reliable estimates.

STEP 4

Let's calculate $n\hat{p}$ : $100 \cdot 0.60 = 60$ .
This is greater than 10, awesome!
Now, let's check $n(1-\hat{p})$ : $100 \cdot (1-0.60) = 100 \cdot 0.40 = 40$ .
This is also greater than 10, fantastic!

STEP 5

Time for $n\hat{p}$ : $40 \cdot 0.25 = 10$ .
Right on the edge, but it works!
Now for $n(1-\hat{p})$ : $40 \cdot (1-0.25) = 40 \cdot 0.75 = 30$ .
Greater than 10, we're good to go!

STEP 6

Let's compute $n\hat{p}$ : $50 \cdot 0.25 = 12.5$ .
Greater than 10, looking good!
Now for $n(1-\hat{p})$ : $50 \cdot (1-0.25) = 50 \cdot 0.75 = 37.5$ .
Also greater than 10, perfect!

STEP 7

Calculating $n\hat{p}$ : $80 \cdot 0.10 = 8$ .
Uh oh, this is less than 10.
We don't even need to check the second condition, since this one already fails.

STEP 8

(a) The sample size is large enough. (b) The sample size is large enough. (c) The sample size is large enough. (d) The sample size is not large enough.

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Studdy Solution

STEP 1

What is this asking? Do we have enough data to use a specific formula about margin of error? Watch out! Don't forget, a large enough sample size depends on both nnn and p^\hat{p}p^​!

STEP 2

1. Check Sample Size

STEP 3

Alright, let's dive into this!To figure out if our sample size is large enough, we need to check if both np^n\hat{p}np^​ and n(1−p^)n(1-\hat{p})n(1−p^​) are greater than or equal to **10**.This magic number **10** ensures we have enough data to make reliable estimates.

STEP 4

STEP 5

Time for np^n\hat{p}np^​: 40⋅0.25=1040 \cdot 0.25 = 1040⋅0.25=10.Right on the edge, but it works!Now for n(1−p^)n(1-\hat{p})n(1−p^​): 40⋅(1−0.25)=40⋅0.75=3040 \cdot (1-0.25) = 40 \cdot 0.75 = 3040⋅(1−0.25)=40⋅0.75=30.Greater than **10**, we're good to go!

STEP 6

Let's compute np^n\hat{p}np^​: 50⋅0.25=12.550 \cdot 0.25 = 12.550⋅0.25=12.5.Greater than **10**, looking good!Now for n(1−p^)n(1-\hat{p})n(1−p^​): 50⋅(1−0.25)=50⋅0.75=37.550 \cdot (1-0.25) = 50 \cdot 0.75 = 37.550⋅(1−0.25)=50⋅0.75=37.5.Also greater than **10**, perfect!

STEP 7

Calculating np^n\hat{p}np^​: 80⋅0.10=880 \cdot 0.10 = 880⋅0.10=8.Uh oh, this is less than **10**.We don't even need to check the second condition, since this one already fails.

STEP 8

(a) The sample size is large enough. (b) The sample size is large enough. (c) The sample size is large enough. (d) The sample size is not large enough.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Do we have enough data to use a specific formula about margin of error? Watch out! Don't forget, a large enough sample size depends on both $n$ and $\hat{p}$ !

Alright, let's dive into this!
To figure out if our sample size is large enough, we need to check if both $n\hat{p}$ and $n(1-\hat{p})$ are greater than or equal to 10.
This magic number 10 ensures we have enough data to make reliable estimates.

Time for $n\hat{p}$ : $40 \cdot 0.25 = 10$ .
Right on the edge, but it works!
Now for $n(1-\hat{p})$ : $40 \cdot (1-0.25) = 40 \cdot 0.75 = 30$ .
Greater than 10, we're good to go!

Let's compute $n\hat{p}$ : $50 \cdot 0.25 = 12.5$ .
Greater than 10, looking good!
Now for $n(1-\hat{p})$ : $50 \cdot (1-0.25) = 50 \cdot 0.75 = 37.5$ .
Also greater than 10, perfect!

Calculating $n\hat{p}$ : $80 \cdot 0.10 = 8$ .
Uh oh, this is less than 10.
We don't even need to check the second condition, since this one already fails.