Math / Data & Statistics

QuestionTest the claim that the mean GPA of night students is smaller than 3.4 at the 0.025 significance level. The null and alternative hypothesis would be: $\begin{array}{cccccc} H_{0}: \mu \geq 3.4 & H_{0}: p=0.85 & H_{0}: p \leq 0.85 & H_{0}: p \geq 0.85 & H_{0}: \mu=3.4 & H_{0}: \mu \leq 3.4 \\ H_{1}: \mu<3.4 & H_{1}: p \neq 0.85 & H_{1}: p>0.85 & H_{1}: p<0.85 & H_{1}: \mu \neq 3.4 & H_{1}: \mu>3.4 \\ & & & & & \end{array}$
The test is: $\qquad$
Based on a sample of 80 people, the sample mean GPA was 3.37 with a standard deviation of 0.06 The test statistic is: $\square$ (to 2 decimals)
The $p$ -value is: $\square$ (to 2 decimals)

Studdy Solution

STEP 1

What is this asking? We need to check if the average GPA of night students is really less than 3.4, and we'll use a special test to do this with some data we collected. Watch out! Don't mix up the null and alternative hypotheses – the null is what we're trying to disprove!
Also, remember that the p-value tells us how likely our results are if the null hypothesis were actually true.

STEP 2

1. Set up the Hypotheses
2. Calculate the Test Statistic
3. Find the p-value

STEP 3

We're testing if the mean GPA of night students is less than 3.4.
This is our alternative hypothesis: $H_1: \mu < 3.4$ .

STEP 4

The null hypothesis is the opposite: $H_0: \mu \geq 3.4$ .
It assumes the mean GPA is greater than or equal to 3.4.

STEP 5

We have a sample size of $n = \textbf{80}$ , a sample mean GPA of $\bar{x} = \textbf{3.37}$ , and a sample standard deviation of $s = \textbf{0.06}$ .
Our hypothesized population mean is $\mu_0 = \textbf{3.4}$ .

STEP 6

Since we're dealing with a sample and we don't know the population standard deviation, we'll use a t-test.
The formula for the t-statistic is: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

STEP 7

Let's plug in our values: $t = \frac{\textbf{3.37} - \textbf{3.4}}{\textbf{0.06} / \sqrt{\textbf{80}}}$ $t = \frac{-0.03}{\textbf{0.06} / 8.944}$ $t = \frac{-0.03}{0.0067}$ $t \approx \textbf{-4.48}$ So, our test statistic is approximately $\textbf{-4.48}$ .

STEP 8

Our test is left-tailed because our alternative hypothesis is $H_1: \mu < 3.4$ .
This means we're looking for the probability of getting a t-statistic as extreme as $\textbf{-4.48}$ or more extreme (i.e., smaller) if the null hypothesis were true.

STEP 9

We'll use a t-distribution table or a calculator with degrees of freedom $df = n - 1 = \textbf{80} - 1 = \textbf{79}$ .
Looking up our test statistic ( $\textbf{-4.48}$ ), we find a p-value that is very small, much less than our significance level of $\alpha = \textbf{0.025}$ .
Since most t-tables don't have values this extreme, we can safely approximate the p-value to be approximately $\textbf{0.00}$ .

STEP 10

The test statistic is approximately $\textbf{-4.48}$ .
The ** $p$ -value** is approximately $\textbf{0.00}$ .
Since our p-value is less than our significance level ( $0.00 < 0.025$ ), we reject the null hypothesis.
There is sufficient evidence to support the claim that the mean GPA of night students is smaller than 3.4.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Set up the Hypotheses
2. Calculate the Test Statistic
3. Find the p-value

STEP 3

We're testing if the mean GPA of night students is less than 3.4.
This is our alternative hypothesis: $H_1: \mu < 3.4$ .

STEP 4

The null hypothesis is the opposite: $H_0: \mu \geq 3.4$ .
It assumes the mean GPA is greater than or equal to 3.4.

STEP 5

We have a sample size of $n = \textbf{80}$ , a sample mean GPA of $\bar{x} = \textbf{3.37}$ , and a sample standard deviation of $s = \textbf{0.06}$ .
Our hypothesized population mean is $\mu_0 = \textbf{3.4}$ .

STEP 6

Since we're dealing with a sample and we don't know the population standard deviation, we'll use a t-test.
The formula for the t-statistic is: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

STEP 7

Let's plug in our values: $t = \frac{\textbf{3.37} - \textbf{3.4}}{\textbf{0.06} / \sqrt{\textbf{80}}}$ $t = \frac{-0.03}{\textbf{0.06} / 8.944}$ $t = \frac{-0.03}{0.0067}$ $t \approx \textbf{-4.48}$ So, our test statistic is approximately $\textbf{-4.48}$ .

STEP 8

Our test is left-tailed because our alternative hypothesis is $H_1: \mu < 3.4$ .
This means we're looking for the probability of getting a t-statistic as extreme as $\textbf{-4.48}$ or more extreme (i.e., smaller) if the null hypothesis were true.

STEP 9

STEP 10

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

STEP 2

1. Set up the Hypotheses2. Calculate the Test Statistic3. Find the *p*-value

STEP 3

We're testing if the mean GPA of night students is **less than** 3.4.This is our **alternative hypothesis**: H1:μ<3.4H_1: \mu < 3.4H1​:μ<3.4.

STEP 4

The **null hypothesis** is the opposite: H0:μ≥3.4H_0: \mu \geq 3.4H0​:μ≥3.4.It assumes the mean GPA is **greater than or equal to** 3.4.

STEP 5

We have a **sample size** of n=80n = \textbf{80}n=80, a **sample mean** GPA of xˉ=3.37\bar{x} = \textbf{3.37}xˉ=3.37, and a **sample standard deviation** of s=0.06s = \textbf{0.06}s=0.06.Our **hypothesized population mean** is μ0=3.4\mu_0 = \textbf{3.4}μ0​=3.4.

STEP 6

Since we're dealing with a sample and we don't know the population standard deviation, we'll use a *t*-test.The formula for the *t*-statistic is: t=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}t=s/n​xˉ−μ0​​

STEP 7

STEP 8

Our test is **left-tailed** because our alternative hypothesis is H1:μ<3.4H_1: \mu < 3.4H1​:μ<3.4.This means we're looking for the probability of getting a *t*-statistic as extreme as -4.48\textbf{-4.48}-4.48 or more extreme (i.e., smaller) if the null hypothesis were true.

STEP 9

STEP 10

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.

1. Set up the Hypotheses
2. Calculate the Test Statistic
3. Find the p-value

We're testing if the mean GPA of night students is less than 3.4.
This is our alternative hypothesis: $H_1: \mu < 3.4$ .

The null hypothesis is the opposite: $H_0: \mu \geq 3.4$ .
It assumes the mean GPA is greater than or equal to 3.4.

We have a sample size of $n = \textbf{80}$ , a sample mean GPA of $\bar{x} = \textbf{3.37}$ , and a sample standard deviation of $s = \textbf{0.06}$ .
Our hypothesized population mean is $\mu_0 = \textbf{3.4}$ .

Since we're dealing with a sample and we don't know the population standard deviation, we'll use a t-test.
The formula for the t-statistic is: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

Our test is left-tailed because our alternative hypothesis is $H_1: \mu < 3.4$ .
This means we're looking for the probability of getting a t-statistic as extreme as $\textbf{-4.48}$ or more extreme (i.e., smaller) if the null hypothesis were true.