Math / Data & Statistics

QuestionThe formula used to compute a large-sample confidence interval for $p$ is $\hat{p} \pm(z \text { critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
What is the appropriate $z$ critical value for each of the following confidence levels? (Round your answers to two decimal places.) (a) $95 \%$ $\square$ (b) $90 \%$ $\square$ (c) $99 \%$ $\square$ (d) $80 \%$ $\square$ (e) $81 \%$ $\square$

Studdy Solution

STEP 1

What is this asking? Find the z critical values for different confidence levels, which are used in calculating confidence intervals. Watch out! Don't mix up confidence levels with alpha values!
Also, remember to round to two decimal places.

STEP 2

1. 95% Confidence Level
2. 90% Confidence Level
3. 99% Confidence Level
4. 80% Confidence Level
5. 81% Confidence Level

STEP 3

Alright, let's kick things off with the 95% confidence level!
This means we want to capture the true population parameter 95% of the time.

STEP 4

To find our z critical value, we need to think about the area in the tails of our normal distribution.
Since 95% is in the middle, we have $(1 - 0.95) = 0.05$ left over for both tails.

STEP 5

Because the normal distribution is symmetric, we split that 0.05 in half, giving us $0.05 / 2 = \textbf{0.025}$ in each tail.

STEP 6

Now, we look up this area in a z-table (or use a calculator) to find the corresponding z-score.
We're looking for the z-score where 0.025 of the area is to the right of it.
This gives us a z critical value of approximately $\textbf{1.96}$ .

STEP 7

On to the next!
For a 90% confidence level, we follow the same process.

STEP 8

We have $(1 - 0.90) = 0.10$ left for the tails, and splitting that in half gives us $0.10 / 2 = \textbf{0.05}$ in each tail.

STEP 9

Looking up this tail area in our z-table gives us a z critical value of approximately $\textbf{1.645}$ , which rounds to $\textbf{1.65}$ .

STEP 10

Time for 99%!
This one's a high confidence level, so our z critical value will be larger.

STEP 11

We have $(1 - 0.99) = 0.01$ for the tails, and $0.01 / 2 = \textbf{0.005}$ for each tail.

STEP 12

The z-table tells us that a tail area of 0.005 corresponds to a z critical value of approximately $\textbf{2.575}$ , which we round up to $\textbf{2.58}$ .

STEP 13

Now for 80%!
We've got $(1 - 0.80) = 0.20$ for both tails, and $0.20 / 2 = \textbf{0.10}$ for each tail.

STEP 14

Our trusty z-table gives us a z critical value of approximately $\textbf{1.28}$ .

STEP 15

Last one, let's do it!
For 81%, we have $(1 - 0.81) = 0.19$ total tail area, and $0.19 / 2 = \textbf{0.095}$ for each tail.

STEP 16

Looking this up gives us a z critical value of approximately $\textbf{1.31}$ .

STEP 17

(a) $1.96$ (b) $1.65$ (c) $2.58$ (d) $1.28$ (e) $1.31$

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Studdy Solution

STEP 1

What is this asking? Find the z critical values for different confidence levels, which are used in calculating confidence intervals. Watch out! Don't mix up confidence levels with alpha values!
Also, remember to round to two decimal places.

STEP 2

1. 95% Confidence Level
2. 90% Confidence Level
3. 99% Confidence Level
4. 80% Confidence Level
5. 81% Confidence Level

STEP 3

Alright, let's kick things off with the 95% confidence level!
This means we want to capture the true population parameter 95% of the time.

STEP 4

To find our z critical value, we need to think about the area in the tails of our normal distribution.
Since 95% is in the middle, we have $(1 - 0.95) = 0.05$ left over for both tails.

STEP 5

Because the normal distribution is symmetric, we split that 0.05 in half, giving us $0.05 / 2 = \textbf{0.025}$ in each tail.

STEP 6

Now, we look up this area in a z-table (or use a calculator) to find the corresponding z-score.
We're looking for the z-score where 0.025 of the area is to the right of it.
This gives us a z critical value of approximately $\textbf{1.96}$ .

STEP 7

On to the next!
For a 90% confidence level, we follow the same process.

STEP 8

We have $(1 - 0.90) = 0.10$ left for the tails, and splitting that in half gives us $0.10 / 2 = \textbf{0.05}$ in each tail.

STEP 9

Looking up this tail area in our z-table gives us a z critical value of approximately $\textbf{1.645}$ , which rounds to $\textbf{1.65}$ .

STEP 10

Time for 99%!
This one's a high confidence level, so our z critical value will be larger.

STEP 11

We have $(1 - 0.99) = 0.01$ for the tails, and $0.01 / 2 = \textbf{0.005}$ for each tail.

STEP 12

The z-table tells us that a tail area of 0.005 corresponds to a z critical value of approximately $\textbf{2.575}$ , which we round up to $\textbf{2.58}$ .

STEP 13

Now for 80%!
We've got $(1 - 0.80) = 0.20$ for both tails, and $0.20 / 2 = \textbf{0.10}$ for each tail.

STEP 14

Our trusty z-table gives us a z critical value of approximately $\textbf{1.28}$ .

STEP 15

Last one, let's do it!
For 81%, we have $(1 - 0.81) = 0.19$ total tail area, and $0.19 / 2 = \textbf{0.095}$ for each tail.

STEP 16

Looking this up gives us a z critical value of approximately $\textbf{1.31}$ .

STEP 17

(a) $1.96$ (b) $1.65$ (c) $2.58$ (d) $1.28$ (e) $1.31$

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Studdy Solution

STEP 1

What is this asking? Find the *z* critical values for different confidence levels, which are used in calculating confidence intervals. Watch out! Don't mix up confidence levels with alpha values!Also, remember to round to two decimal places.

STEP 2

1. 95% Confidence Level2. 90% Confidence Level3. 99% Confidence Level4. 80% Confidence Level5. 81% Confidence Level

STEP 3

Alright, let's **kick things off** with the 95% confidence level!This means we want to capture the true population parameter 95% of the time.

STEP 4

To find our *z* critical value, we need to think about the area in the tails of our normal distribution.Since 95% is in the middle, we have (1−0.95)=0.05(1 - 0.95) = 0.05(1−0.95)=0.05 left over for *both* tails.

STEP 5

Because the normal distribution is symmetric, we **split that 0.05 in half**, giving us 0.05/2=0.0250.05 / 2 = \textbf{0.025}0.05/2=0.025 in each tail.

STEP 6

Now, we look up this area in a *z*-table (or use a calculator) to find the corresponding *z*-score.We're looking for the *z*-score where 0.025 of the area is to the right of it.This gives us a *z* critical value of approximately 1.96\textbf{1.96}1.96.

STEP 7

On to the next!For a 90% confidence level, we follow the same process.

STEP 8

We have (1−0.90)=0.10(1 - 0.90) = 0.10(1−0.90)=0.10 left for the tails, and splitting that in half gives us 0.10/2=0.050.10 / 2 = \textbf{0.05}0.10/2=0.05 in each tail.

STEP 9

Looking up this tail area in our *z*-table gives us a *z* critical value of approximately 1.645\textbf{1.645}1.645, which rounds to 1.65\textbf{1.65}1.65.

STEP 10

Time for 99%!This one's a high confidence level, so our *z* critical value will be larger.

STEP 11

We have (1−0.99)=0.01(1 - 0.99) = 0.01(1−0.99)=0.01 for the tails, and 0.01/2=0.0050.01 / 2 = \textbf{0.005}0.01/2=0.005 for each tail.

STEP 12

The *z*-table tells us that a tail area of 0.005 corresponds to a *z* critical value of approximately 2.575\textbf{2.575}2.575, which we round up to 2.58\textbf{2.58}2.58.

STEP 13

Now for 80%!We've got (1−0.80)=0.20(1 - 0.80) = 0.20(1−0.80)=0.20 for both tails, and 0.20/2=0.100.20 / 2 = \textbf{0.10}0.20/2=0.10 for each tail.

STEP 14

Our trusty *z*-table gives us a *z* critical value of approximately 1.28\textbf{1.28}1.28.

STEP 15

Last one, let's do it!For 81%, we have (1−0.81)=0.19(1 - 0.81) = 0.19(1−0.81)=0.19 total tail area, and 0.19/2=0.0950.19 / 2 = \textbf{0.095}0.19/2=0.095 for each tail.

STEP 16

Looking this up gives us a *z* critical value of approximately 1.31\textbf{1.31}1.31.

STEP 17

(a) 1.961.961.96 (b) 1.651.651.65 (c) 2.582.582.58 (d) 1.281.281.28 (e) 1.311.311.31

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What is this asking? Find the z critical values for different confidence levels, which are used in calculating confidence intervals. Watch out! Don't mix up confidence levels with alpha values!
Also, remember to round to two decimal places.

1. 95% Confidence Level
2. 90% Confidence Level
3. 99% Confidence Level
4. 80% Confidence Level
5. 81% Confidence Level

Alright, let's kick things off with the 95% confidence level!
This means we want to capture the true population parameter 95% of the time.

To find our z critical value, we need to think about the area in the tails of our normal distribution.
Since 95% is in the middle, we have $(1 - 0.95) = 0.05$ left over for both tails.

Because the normal distribution is symmetric, we split that 0.05 in half, giving us $0.05 / 2 = \textbf{0.025}$ in each tail.

Now, we look up this area in a z-table (or use a calculator) to find the corresponding z-score.
We're looking for the z-score where 0.025 of the area is to the right of it.
This gives us a z critical value of approximately $\textbf{1.96}$ .

On to the next!
For a 90% confidence level, we follow the same process.

We have $(1 - 0.90) = 0.10$ left for the tails, and splitting that in half gives us $0.10 / 2 = \textbf{0.05}$ in each tail.

Looking up this tail area in our z-table gives us a z critical value of approximately $\textbf{1.645}$ , which rounds to $\textbf{1.65}$ .

Time for 99%!
This one's a high confidence level, so our z critical value will be larger.

We have $(1 - 0.99) = 0.01$ for the tails, and $0.01 / 2 = \textbf{0.005}$ for each tail.

The z-table tells us that a tail area of 0.005 corresponds to a z critical value of approximately $\textbf{2.575}$ , which we round up to $\textbf{2.58}$ .

Now for 80%!
We've got $(1 - 0.80) = 0.20$ for both tails, and $0.20 / 2 = \textbf{0.10}$ for each tail.

Our trusty z-table gives us a z critical value of approximately $\textbf{1.28}$ .

Last one, let's do it!
For 81%, we have $(1 - 0.81) = 0.19$ total tail area, and $0.19 / 2 = \textbf{0.095}$ for each tail.

Looking this up gives us a z critical value of approximately $\textbf{1.31}$ .

(a) $1.96$ (b) $1.65$ (c) $2.58$ (d) $1.28$ (e) $1.31$