Math / Data & Statistics

QuestionDoes the average Presbyterian donate more than the average Catholic in church on Sundays? The 58 randomly observed members of the Presbyterian church donated an average of $\$ 27$ with a standard deviation of \ $13. The 55 randomly observed members of the Catholic church donated an average of \$26 with a standard deviation of$ \ $7$ . What can be concluded at the $\alpha=0.10$ level of significance? a. For this study, we should use Select an answer b. The null and alternative hypotheses would be: $H_{0}$ : Select an answer Select an answer Select an answer (2) (please enter a decimal) $H_{1}$ : Select an answer Select an answer Select an answer (Please enter a decimal) c. The test statistic ? : = $\square$ (please show your answer to 3 decimal places.) d. The $p$ -value $=$ $\square$ (Please show your answer to 4 decimal places.) e. The $p$ -value is ? $\alpha$ f. Based on this, we should Select an answer the null hypothesis. g. Thus, the final conclusion is that ... The results are statistically significant at $\alpha=0.10$ , so there is sufficient evidence to conclude that the mean donation for the 58 Presbyterians that were observed is more than the mean donation for the 55 Catholics that were observed. The results are statistically significant at $\alpha=0.10$ , so there is sufficient evidence to conclude that the population mean amount of money that Presbyterians donate is more than the population mean amount of money that Catholics donate. The results are statistically insignificant at $\alpha=0.10$ , so there is insufficient evidence to conclude that the population mean amount of money that Presbyterians donate is more than the population mean amount of money that Catholics donate. The results are statistically insignificant at $\alpha=0.10$ , so there is statistically significant evidence to conclude that the population mean amount of money that Presbyterians donate is equal to the population mean amount of money that Catholics donate.

Studdy Solution

STEP 1

What is this asking? We want to find out if Presbyterians donate more money on average than Catholics, based on samples from both groups. Watch out! Don't mix up the sample data with the population parameters.
We're using the samples to infer something about the populations!

STEP 2

1. Set up the hypothesis test
2. Calculate the test statistic
3. Find the p-value
4. Make a decision

STEP 3

Our null hypothesis ( $H_0$ ) is that there's no difference in average donations between the two groups.
Mathematically, this means: $H_0: \mu_P - \mu_C \le 0$ where $\mu_P$ is the population mean donation of Presbyterians and $\mu_C$ is the population mean donation of Catholics.
In other words, the difference is less than or equal to zero.

STEP 4

Our alternative hypothesis ( $H_1$ ) is what we're trying to find evidence for.
In this case, it's that Presbyterians donate more on average: $H_1: \mu_P - \mu_C > 0$ Meaning the difference is greater than zero.
This is a right-tailed test!

STEP 5

The significance level, $\alpha$ , is 0.10.
This is our cutoff for deciding if the results are statistically significant.

STEP 6

Since we have two independent samples and we're comparing means, we'll use a two-sample t-test.

STEP 7

The formula for the two-sample t-test is: $t = \frac{(\bar{x}_P - \bar{x}_C) - (\mu_P - \mu_C)}{\sqrt{\frac{s_P^2}{n_P} + \frac{s_C^2}{n_C}}}$ Here, $\bar{x}_P = \$27$ and $\bar{x}_C = \$26$ are the sample means, $s_P = \$13$ and $s_C = \$7$ are the sample standard deviations, and $n_P = 58$ and $n_C = 55$ are the sample sizes.
Under the null hypothesis, $(\mu_P - \mu_C) = 0$ .
Plugging in the values: $t = \frac{(\$27 - \$26) - 0}{\sqrt{\frac{(\$13)^2}{58} + \frac{(\$7)^2}{55}}} = \frac{\$1}{\sqrt{\frac{\$169}{58} + \frac{\$49}{55}}} = \frac{\$1}{\sqrt{\$2.914 + \$0.891}} \approx \frac{\$1}{\sqrt{\$3.805}} \approx \frac{\$1}{\$1.951} \approx 0.513$ So, our test statistic is approximately 0.513.

STEP 8

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one we calculated, assuming the null hypothesis is true.

STEP 9

Since this is a right-tailed test, we want the probability of getting a t-value greater than or equal to 0.513.
Using a t-distribution calculator or table with the appropriate degrees of freedom (which can be approximated), we find the p-value is approximately 0.3050.

STEP 10

Our p-value (0.3050) is greater than our significance level (0.10).

STEP 11

Since the p-value is greater than alpha, we fail to reject the null hypothesis.

STEP 12

a. Two-Sample t-test b. $H_0: \mu_P - \mu_C \le 0$ , $H_1: \mu_P - \mu_C > 0$ c. $t \approx 0.513$ d. $p \approx 0.3050$ e. $p > \alpha$ f. Fail to reject g. The results are statistically insignificant at $\alpha = 0.10$ , so there is insufficient evidence to conclude that the population mean amount of money that Presbyterians donate is more than the population mean amount of money that Catholics donate.

Was this helpful?

Studdy Solution

STEP 1

What is this asking? We want to find out if Presbyterians donate more money on average than Catholics, based on samples from both groups. Watch out! Don't mix up the sample data with the population parameters.
We're using the samples to infer something about the populations!

STEP 2

1. Set up the hypothesis test
2. Calculate the test statistic
3. Find the p-value
4. Make a decision

STEP 3

STEP 4

Our alternative hypothesis ( $H_1$ ) is what we're trying to find evidence for.
In this case, it's that Presbyterians donate more on average: $H_1: \mu_P - \mu_C > 0$ Meaning the difference is greater than zero.
This is a right-tailed test!

STEP 5

The significance level, $\alpha$ , is 0.10.
This is our cutoff for deciding if the results are statistically significant.

STEP 6

Since we have two independent samples and we're comparing means, we'll use a two-sample t-test.

STEP 7

STEP 8

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one we calculated, assuming the null hypothesis is true.

STEP 9

Since this is a right-tailed test, we want the probability of getting a t-value greater than or equal to 0.513.
Using a t-distribution calculator or table with the appropriate degrees of freedom (which can be approximated), we find the p-value is approximately 0.3050.

STEP 10

Our p-value (0.3050) is greater than our significance level (0.10).

STEP 11

Since the p-value is greater than alpha, we fail to reject the null hypothesis.

STEP 12

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.

Studdy Solution

STEP 1

What is this asking? We want to find out if Presbyterians donate more money on average than Catholics, based on samples from both groups. Watch out! Don't mix up the sample data with the population parameters.We're using the samples to *infer* something about the populations!

STEP 2

1. Set up the hypothesis test2. Calculate the test statistic3. Find the p-value4. Make a decision

STEP 3

STEP 4

Our **alternative hypothesis** (H1H_1H1​) is what we're trying to find evidence *for*.In this case, it's that Presbyterians donate *more* on average: H1:μP−μC>0H_1: \mu_P - \mu_C > 0H1​:μP​−μC​>0 Meaning the difference is greater than zero.This is a *right-tailed test*!

STEP 5

The significance level, α\alphaα, is **0.10**.This is our cutoff for deciding if the results are statistically significant.

STEP 6

Since we have two independent samples and we're comparing means, we'll use a **two-sample t-test**.

STEP 7

STEP 8

The **p-value** is the probability of observing a test statistic as extreme as, or more extreme than, the one we calculated, *assuming the null hypothesis is true*.

STEP 9

Since this is a right-tailed test, we want the probability of getting a t-value greater than or equal to **0.513**.Using a t-distribution calculator or table with the appropriate degrees of freedom (which can be approximated), we find the p-value is approximately **0.3050**.

STEP 10

Our **p-value (0.3050)** is *greater* than our **significance level (0.10)**.

STEP 11

Since the p-value is greater than alpha, we **fail to reject the null hypothesis**.

STEP 12

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? We want to find out if Presbyterians donate more money on average than Catholics, based on samples from both groups. Watch out! Don't mix up the sample data with the population parameters.
We're using the samples to infer something about the populations!

1. Set up the hypothesis test
2. Calculate the test statistic
3. Find the p-value
4. Make a decision

Our alternative hypothesis ( $H_1$ ) is what we're trying to find evidence for.
In this case, it's that Presbyterians donate more on average: $H_1: \mu_P - \mu_C > 0$ Meaning the difference is greater than zero.
This is a right-tailed test!

The significance level, $\alpha$ , is 0.10.
This is our cutoff for deciding if the results are statistically significant.

Since we have two independent samples and we're comparing means, we'll use a two-sample t-test.

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one we calculated, assuming the null hypothesis is true.

Since this is a right-tailed test, we want the probability of getting a t-value greater than or equal to 0.513.
Using a t-distribution calculator or table with the appropriate degrees of freedom (which can be approximated), we find the p-value is approximately 0.3050.

Our p-value (0.3050) is greater than our significance level (0.10).

Since the p-value is greater than alpha, we fail to reject the null hypothesis.