Math / Data & Statistics

QuestionAccording to the Federal Trade Commission report on consumer fraud and identity theft, $23 \%$ of all complaints in 2007 were for identity theft. This year, a certain state kept track of the number of its 1415 complaints were for identity theft. They want to know if the data provide enough evidence to show that this state had a higher proportion of identity theft than $23 \%$ ? State the random variable, population parameter, and hypotheses. a. The symbol for the random variable involved in this problem is b. The wording for the random variable in context is as follows: Select an answer c. The symbol for the parameter involved in this problem is $\square$ $\square$ d. The wording for the parameter in context is as follows: Select an answer e. Fill in the correct null and alternative hypotheses. $\begin{array}{l} H_{0}: ? \vee ? v \square \\ H_{A}: ? \vee ? \vee \square \end{array}$ f. A Type I error in the context of this problem would be $\square$ g. A Type II error in the context of this problem would be
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Studdy Solution

STEP 1

What is this asking? Does this state have a higher percentage of identity theft complaints than the national average of $\$23\%$ reported in 2007? Watch out! Don't mix up the sample proportion from the state with the population parameter (the national average).
Also, be careful not to confuse Type I and Type II errors!

STEP 2

1. Define the random variable and parameter.
2. State the hypotheses.
3. Describe Type I and Type II errors.

STEP 3

The random variable ( $\hat{p}$ ) is the proportion of identity theft complaints in the sample from this state.
It's what we're measuring and it can change!

STEP 4

The population parameter ( $p$ ) is the true proportion of identity theft complaints nationally, which is $\$23\%$ or $0.23$ .
This is what we're comparing our sample against.

STEP 5

The null hypothesis ( $H_0$ ) is what we assume is true unless we find strong evidence against it.
Here, it's that the state's identity theft proportion is not higher than the national average.
So, $H_0: p \le 0.23$ .

STEP 6

The alternative hypothesis ( $H_A$ ) is what we're trying to find evidence for.
In this case, it's that the state's proportion is higher than the national average.
So, $H_A: p > 0.23$ .

STEP 7

A Type I error is when we reject the null hypothesis when it's actually true.
Here, that would mean concluding the state has a higher identity theft proportion when it really doesn't.
It's like a false alarm!

STEP 8

A Type II error is when we fail to reject the null hypothesis when it's actually false.
Here, that would mean concluding the state's identity theft proportion isn't higher when it actually is.
It's like missing the real problem!

STEP 9

a. $\hat{p}$ b. Proportion of identity theft complaints in the sample from this state. c. $p$ d. True proportion of identity theft complaints nationally. e. $H_0: p \le 0.23$ , $H_A: p > 0.23$ f. Concluding the state has a higher proportion of identity theft complaints when it really doesn't. g. Concluding the state does not have a higher proportion of identity theft complaints when it actually does.

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Studdy Solution

STEP 1

STEP 2

1. Define the random variable and parameter.
2. State the hypotheses.
3. Describe Type I and Type II errors.

STEP 3

The random variable ( $\hat{p}$ ) is the proportion of identity theft complaints in the sample from this state.
It's what we're measuring and it can change!

STEP 4

The population parameter ( $p$ ) is the true proportion of identity theft complaints nationally, which is $\$23\%$ or $0.23$ .
This is what we're comparing our sample against.

STEP 5

The null hypothesis ( $H_0$ ) is what we assume is true unless we find strong evidence against it.
Here, it's that the state's identity theft proportion is not higher than the national average.
So, $H_0: p \le 0.23$ .

STEP 6

The alternative hypothesis ( $H_A$ ) is what we're trying to find evidence for.
In this case, it's that the state's proportion is higher than the national average.
So, $H_A: p > 0.23$ .

STEP 7

A Type I error is when we reject the null hypothesis when it's actually true.
Here, that would mean concluding the state has a higher identity theft proportion when it really doesn't.
It's like a false alarm!

STEP 8

A Type II error is when we fail to reject the null hypothesis when it's actually false.
Here, that would mean concluding the state's identity theft proportion isn't higher when it actually is.
It's like missing the real problem!

STEP 9

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Studdy Solution

STEP 1

STEP 2

1. Define the random variable and parameter.2. State the hypotheses.3. Describe Type I and Type II errors.

STEP 3

The **random variable** (p^\hat{p}p^​) is the *proportion* of identity theft complaints in the sample from this state.It's what we're *measuring* and it can change!

STEP 4

The **population parameter** (ppp) is the *true proportion* of identity theft complaints nationally, which is $23%\$23\%$23% or 0.230.230.23.This is what we're comparing our sample against.

STEP 5

The **null hypothesis** (H0H_0H0​) is what we *assume* is true unless we find strong evidence against it.Here, it's that the state's identity theft proportion is *not* higher than the national average.So, H0:p≤0.23H_0: p \le 0.23H0​:p≤0.23.

STEP 6

The **alternative hypothesis** (HAH_AHA​) is what we're trying to find evidence *for*.In this case, it's that the state's proportion *is* higher than the national average.So, HA:p>0.23H_A: p > 0.23HA​:p>0.23.

STEP 7

A **Type I error** is when we *reject* the null hypothesis when it's actually *true*.Here, that would mean concluding the state has a higher identity theft proportion when it really doesn't.It's like a *false alarm*!

STEP 8

A **Type II error** is when we *fail to reject* the null hypothesis when it's actually *false*.Here, that would mean concluding the state's identity theft proportion isn't higher when it actually is.It's like *missing* the real problem!

STEP 9

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1. Define the random variable and parameter.
2. State the hypotheses.
3. Describe Type I and Type II errors.

The random variable ( $\hat{p}$ ) is the proportion of identity theft complaints in the sample from this state.
It's what we're measuring and it can change!

The population parameter ( $p$ ) is the true proportion of identity theft complaints nationally, which is $\$23\%$ or $0.23$ .
This is what we're comparing our sample against.

The null hypothesis ( $H_0$ ) is what we assume is true unless we find strong evidence against it.
Here, it's that the state's identity theft proportion is not higher than the national average.
So, $H_0: p \le 0.23$ .

The alternative hypothesis ( $H_A$ ) is what we're trying to find evidence for.
In this case, it's that the state's proportion is higher than the national average.
So, $H_A: p > 0.23$ .

A Type I error is when we reject the null hypothesis when it's actually true.
Here, that would mean concluding the state has a higher identity theft proportion when it really doesn't.
It's like a false alarm!

A Type II error is when we fail to reject the null hypothesis when it's actually false.
Here, that would mean concluding the state's identity theft proportion isn't higher when it actually is.
It's like missing the real problem!