Math / Data & Statistics

QuestionBack in 1993, a study found 52 in a sample of 503 European women were mathematicians. A similar study, conducted in 2005, found 88 in a sample of 495 European women were mathematicians. Let $p_1$ , $p_2$ represent the proportion of European woman mathematicians in 1993 and in 2005 respectively. At significance level 0.002, is this good evidence showing that the proportion of women mathematicians in Europe has changed in 12 years since 1993?
A) State the Null and the Alternate Hypotheses: $H_0: p_1 = p_2$ ( $dp = 0$ ) $H_1: p_1 \ne p_2$ ( $dp \ne 0$ )
B) Is this a (left-tail, right-tail, one-tail, two-tail) test? two-tail
C) Using 4 decimal places for $p_1$ and $p_2$ , find the value of $d\hat{p}$ . $d\hat{p} =$ ____ Note: $d\hat{p}$ must be in 4 decimal places.
D) Find the value of $\hat{p}_c$ . $\hat{p}_c = 1.1403$ Round your answer to 4 decimal places.
E) Find $\sigma_{d\hat{p}}$ , the standard error of $d\hat{p}$ 's: $\sigma_{d\hat{p}} =$ ____ Round your answer to 4 decimal places.
F) Calculate the z-score of your sampling difference in proportion. $z_{d\hat{p}} =$ Enter your answer in 2 decimal places.
G) Using z-score from part (F), what is the p-value of this test? p-value $=$

Studdy Solution

STEP 1

What is this asking? We're comparing the proportion of female mathematicians in Europe between 1993 and 2005 to see if there's been a significant change! Watch out! Don't mix up $p_1$ and $p_2$ - $p_1$ is for 1993, and $p_2$ is for 2005.
Keep your years straight!

STEP 2

1. Identify hypotheses and test type
2. Calculate sample proportions
3. Find the pooled proportion
4. Calculate the standard error
5. Compute the z-score
6. Determine the p-value

STEP 3

Alright, let's kick this off by looking at our hypotheses!
The question has already given us the correct null and alternate hypotheses:
$H_0: p_1 = p_2 \text{ (dp = 0)}$ H_1: p_1 \neq p_2 \text{ (dp ≠ 0)}
This is saying that our null hypothesis assumes no change in the proportion of women mathematicians, while our alternate hypothesis suggests there is a change (either increase or decrease).

STEP 4

Now, what kind of test are we dealing with?
The question tells us it's a two-tail test.
This makes sense because we're looking for any change, not just an increase or decrease.

STEP 5

Let's crunch some numbers!
We need to find $p_1$ and $p_2$ .
Remember, $p_1$ is for 1993 and $p_2$ is for 2005.
For 1993: $p_1 = \frac{52}{503} = 0.1034$
For 2005: $p_2 = \frac{88}{495} = 0.1778$

STEP 6

Now we can find dp (the difference in proportions): $dp = p_2 - p_1 = 0.1778 - 0.1034 = 0.0744$
Rounding to 4 decimal places as requested: $dp = 0.0744$

STEP 7

The pooled proportion $\bar{p}$ is already given to us in the problem: $\bar{p} = 0.1403$
This value represents the overall proportion if we combine both samples.
It's a weighted average of $p_1$ and $p_2$ .

STEP 8

Now, let's calculate the standard error of dp.
We'll use this formula:
$\sigma_{dp} = \sqrt{\bar{p}(1-\bar{p})(\frac{1}{n_1} + \frac{1}{n_2})}$
Where $n_1 = 503$ and $n_2 = 495$

STEP 9

Let's plug in our values:
$\sigma_{dp} = \sqrt{0.1403(1-0.1403)(\frac{1}{503} + \frac{1}{495})}$

STEP 10

Time to crunch these numbers:
$\sigma_{dp} = \sqrt{0.1403 \cdot 0.8597 \cdot (0.001988 + 0.002020)}$ $\sigma_{dp} = \sqrt{0.1206 \cdot 0.004008}$ $\sigma_{dp} = \sqrt{0.000483} = 0.0220$
Rounding to 4 decimal places: $\sigma_{dp} = 0.0220$

STEP 11

Now we're ready to calculate our z-score!
Here's the formula:
$z_{dp} = \frac{dp - 0}{\sigma_{dp}}$
We use 0 in the numerator because our null hypothesis assumes no difference.

STEP 12

Let's plug in our values:
$z_{dp} = \frac{0.0744 - 0}{0.0220} = \frac{0.0744}{0.0220} = 3.38$
Rounding to 2 decimal places as requested: $z_{dp} = 3.38$

STEP 13

To find the p-value, we need to use a z-table or a calculator.
Since this is a two-tailed test, we'll find the area under both tails of the normal distribution.

STEP 14

Using a z-table or calculator, we find that the area to the right of z = 3.38 is approximately 0.000362.

STEP 15

Since this is a two-tailed test, we double this value:
p-value = 2 * 0.000362 = 0.000724
Rounding to 4 decimal places: p-value = 0.0007

STEP 16

A) The null hypothesis is $H_0: p_1 = p_2$ (dp = 0), and the alternative hypothesis is $H_1: p_1 \neq p_2$ (dp ≠ 0).
B) This is a two-tail test.
C) The value of dp is 0.0744.
D) The value of $\bar{p}$ is 0.1403.
E) The standard error of dp (σdp) is 0.0220.
F) The z-score of the sampling difference in proportion is 3.38.
G) The p-value of this test is 0.0007.

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Studdy Solution

STEP 1

What is this asking? We're comparing the proportion of female mathematicians in Europe between 1993 and 2005 to see if there's been a significant change! Watch out! Don't mix up $p_1$ and $p_2$ - $p_1$ is for 1993, and $p_2$ is for 2005.
Keep your years straight!

STEP 2

1. Identify hypotheses and test type
2. Calculate sample proportions
3. Find the pooled proportion
4. Calculate the standard error
5. Compute the z-score
6. Determine the p-value

STEP 3

STEP 4

Now, what kind of test are we dealing with?
The question tells us it's a two-tail test.
This makes sense because we're looking for any change, not just an increase or decrease.

STEP 5

Let's crunch some numbers!
We need to find $p_1$ and $p_2$ .
Remember, $p_1$ is for 1993 and $p_2$ is for 2005.
For 1993: $p_1 = \frac{52}{503} = 0.1034$
For 2005: $p_2 = \frac{88}{495} = 0.1778$

STEP 6

Now we can find dp (the difference in proportions): $dp = p_2 - p_1 = 0.1778 - 0.1034 = 0.0744$
Rounding to 4 decimal places as requested: $dp = 0.0744$

STEP 7

The pooled proportion $\bar{p}$ is already given to us in the problem: $\bar{p} = 0.1403$
This value represents the overall proportion if we combine both samples.
It's a weighted average of $p_1$ and $p_2$ .

STEP 8

Now, let's calculate the standard error of dp.
We'll use this formula:
$\sigma_{dp} = \sqrt{\bar{p}(1-\bar{p})(\frac{1}{n_1} + \frac{1}{n_2})}$
Where $n_1 = 503$ and $n_2 = 495$

STEP 9

Let's plug in our values:
$\sigma_{dp} = \sqrt{0.1403(1-0.1403)(\frac{1}{503} + \frac{1}{495})}$

STEP 10

Time to crunch these numbers:
$\sigma_{dp} = \sqrt{0.1403 \cdot 0.8597 \cdot (0.001988 + 0.002020)}$ $\sigma_{dp} = \sqrt{0.1206 \cdot 0.004008}$ $\sigma_{dp} = \sqrt{0.000483} = 0.0220$
Rounding to 4 decimal places: $\sigma_{dp} = 0.0220$

STEP 11

Now we're ready to calculate our z-score!
Here's the formula:
$z_{dp} = \frac{dp - 0}{\sigma_{dp}}$
We use 0 in the numerator because our null hypothesis assumes no difference.

STEP 12

Let's plug in our values:
$z_{dp} = \frac{0.0744 - 0}{0.0220} = \frac{0.0744}{0.0220} = 3.38$
Rounding to 2 decimal places as requested: $z_{dp} = 3.38$

STEP 13

To find the p-value, we need to use a z-table or a calculator.
Since this is a two-tailed test, we'll find the area under both tails of the normal distribution.

STEP 14

Using a z-table or calculator, we find that the area to the right of z = 3.38 is approximately 0.000362.

STEP 15

Since this is a two-tailed test, we double this value:
p-value = 2 * 0.000362 = 0.000724
Rounding to 4 decimal places: p-value = 0.0007

STEP 16

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Studdy Solution

STEP 1

What is this asking? We're comparing the proportion of female mathematicians in Europe between 1993 and 2005 to see if there's been a significant change! Watch out! Don't mix up p1p_1p1​ and p2p_2p2​ - p1p_1p1​ is for 1993, and p2p_2p2​ is for 2005.Keep your years straight!

STEP 2

1. Identify hypotheses and test type2. Calculate sample proportions3. Find the pooled proportion4. Calculate the standard error5. Compute the z-score6. Determine the p-value

STEP 3

STEP 4

Now, what kind of test are we dealing with?The question tells us it's a **two-tail test**.This makes sense because we're looking for any change, not just an increase or decrease.

STEP 5

Let's crunch some numbers!We need to find p1p_1p1​ and p2p_2p2​.Remember, p1p_1p1​ is for 1993 and p2p_2p2​ is for 2005.For 1993: p1=52503=0.1034p_1 = \frac{52}{503} = 0.1034p1​=50352​=0.1034For 2005: p2=88495=0.1778p_2 = \frac{88}{495} = 0.1778p2​=49588​=0.1778

STEP 6

Now we can find dp (the difference in proportions): dp=p2−p1=0.1778−0.1034=0.0744dp = p_2 - p_1 = 0.1778 - 0.1034 = 0.0744dp=p2​−p1​=0.1778−0.1034=0.0744Rounding to 4 decimal places as requested: dp=0.0744dp = 0.0744dp=0.0744

STEP 7

The pooled proportion pˉ\bar{p}pˉ​ is already given to us in the problem: pˉ=0.1403\bar{p} = 0.1403pˉ​=0.1403This value represents the overall proportion if we combine both samples.It's a weighted average of p1p_1p1​ and p2p_2p2​.

STEP 8

Now, let's calculate the standard error of dp.We'll use this formula:σdp=pˉ(1−pˉ)(1n1+1n2)\sigma_{dp} = \sqrt{\bar{p}(1-\bar{p})(\frac{1}{n_1} + \frac{1}{n_2})}σdp​=pˉ​(1−pˉ​)(n1​1​+n2​1​)​Where n1=503n_1 = 503n1​=503 and n2=495n_2 = 495n2​=495

STEP 9

Let's plug in our values:σdp=0.1403(1−0.1403)(1503+1495)\sigma_{dp} = \sqrt{0.1403(1-0.1403)(\frac{1}{503} + \frac{1}{495})}σdp​=0.1403(1−0.1403)(5031​+4951​)​

STEP 10

STEP 11

Now we're ready to calculate our z-score!Here's the formula:zdp=dp−0σdpz_{dp} = \frac{dp - 0}{\sigma_{dp}}zdp​=σdp​dp−0​We use 0 in the numerator because our null hypothesis assumes no difference.

STEP 12

Let's plug in our values:zdp=0.0744−00.0220=0.07440.0220=3.38z_{dp} = \frac{0.0744 - 0}{0.0220} = \frac{0.0744}{0.0220} = 3.38zdp​=0.02200.0744−0​=0.02200.0744​=3.38Rounding to 2 decimal places as requested: zdp=3.38z_{dp} = 3.38zdp​=3.38

STEP 13

To find the p-value, we need to use a z-table or a calculator.Since this is a two-tailed test, we'll find the area under both tails of the normal distribution.

STEP 14

Using a z-table or calculator, we find that the area to the right of z = 3.38 is approximately 0.000362.

STEP 15

Since this is a two-tailed test, we double this value:p-value = 2 * 0.000362 = 0.000724Rounding to 4 decimal places: p-value = 0.0007

STEP 16

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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? We're comparing the proportion of female mathematicians in Europe between 1993 and 2005 to see if there's been a significant change! Watch out! Don't mix up $p_1$ and $p_2$ - $p_1$ is for 1993, and $p_2$ is for 2005.
Keep your years straight!

1. Identify hypotheses and test type
2. Calculate sample proportions
3. Find the pooled proportion
4. Calculate the standard error
5. Compute the z-score
6. Determine the p-value

Now, what kind of test are we dealing with?
The question tells us it's a two-tail test.
This makes sense because we're looking for any change, not just an increase or decrease.

Let's crunch some numbers!
We need to find $p_1$ and $p_2$ .
Remember, $p_1$ is for 1993 and $p_2$ is for 2005.
For 1993: $p_1 = \frac{52}{503} = 0.1034$
For 2005: $p_2 = \frac{88}{495} = 0.1778$

Now we can find dp (the difference in proportions): $dp = p_2 - p_1 = 0.1778 - 0.1034 = 0.0744$
Rounding to 4 decimal places as requested: $dp = 0.0744$

The pooled proportion $\bar{p}$ is already given to us in the problem: $\bar{p} = 0.1403$
This value represents the overall proportion if we combine both samples.
It's a weighted average of $p_1$ and $p_2$ .

Now, let's calculate the standard error of dp.
We'll use this formula:
$\sigma_{dp} = \sqrt{\bar{p}(1-\bar{p})(\frac{1}{n_1} + \frac{1}{n_2})}$
Where $n_1 = 503$ and $n_2 = 495$

Let's plug in our values:
$\sigma_{dp} = \sqrt{0.1403(1-0.1403)(\frac{1}{503} + \frac{1}{495})}$

Now we're ready to calculate our z-score!
Here's the formula:
$z_{dp} = \frac{dp - 0}{\sigma_{dp}}$
We use 0 in the numerator because our null hypothesis assumes no difference.

Let's plug in our values:
$z_{dp} = \frac{0.0744 - 0}{0.0220} = \frac{0.0744}{0.0220} = 3.38$
Rounding to 2 decimal places as requested: $z_{dp} = 3.38$

To find the p-value, we need to use a z-table or a calculator.
Since this is a two-tailed test, we'll find the area under both tails of the normal distribution.

Since this is a two-tailed test, we double this value:
p-value = 2 * 0.000362 = 0.000724
Rounding to 4 decimal places: p-value = 0.0007