Math / Data & Statistics

Question $H_0: \mu = 20$ $H_a: \mu < 20$ Use the following information: $n = 48$ , $\overline{X} = 18$ , and $\sigma = 4.6$ To find the test statistic (Step 2).

Studdy Solution

STEP 1

What is this asking? We're checking if the true average is less than 20, given a sample average of 18 from 48 data points and a population standard deviation of 4.6. Watch out! Don't mix up the sample and population standard deviations!
We're given the population standard deviation here, so we'll use a z-test, not a t-test.

STEP 2

1. Set up the z-test
2. Calculate the test statistic

STEP 3

Alright, so we're dealing with a hypothesis test here!
Our null hypothesis $H_0$ says the population mean $\mu$ is 20.
But we suspect it might actually be less than 20, and that's our alternative hypothesis $H_a$ .

STEP 4

Since we know the population standard deviation $\sigma = 4.6$ , we're going to use a z-test.
The z-test formula is: $z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$ .
This formula tells us how far our sample mean $\overline{X}$ is from the hypothesized population mean $\mu$ , measured in terms of standard errors.

STEP 5

Let's plug in what we know.
We have $\overline{X} = 18$ , $\mu = 20$ (from our null hypothesis), $\sigma = 4.6$ , and $n = 48$ .

STEP 6

Substituting the values into our formula, we get: $z = \frac{18 - 20}{\frac{4.6}{\sqrt{48}}}$ .

STEP 7

Let's simplify the denominator first: $\frac{4.6}{\sqrt{48}} \approx \frac{4.6}{6.928} \approx 0.664$ .
So, our standard error is approximately 0.664.

STEP 8

Now, let's compute the numerator: $18 - 20 = -2$ .
This tells us our sample mean is 2 units below our hypothesized mean.

STEP 9

Finally, divide the numerator by the denominator: $z = \frac{-2}{0.664} \approx -3.01$ .
So, our test statistic is approximately $-3.01$ .

STEP 10

Our test statistic is approximately $-3.01$ .
This means our sample mean is about 3 standard deviations below the hypothesized population mean.

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Question $H_0: \mu = 20$ $H_a: \mu < 20$ Use the following information: $n = 48$ , $\overline{X} = 18$ , and $\sigma = 4.6$ To find the test statistic (Step 2).

Studdy Solution

STEP 1

STEP 2

1. Set up the z-test
2. Calculate the test statistic

STEP 3

Alright, so we're dealing with a hypothesis test here!
Our null hypothesis $H_0$ says the population mean $\mu$ is 20.
But we suspect it might actually be less than 20, and that's our alternative hypothesis $H_a$ .

STEP 4

STEP 5

Let's plug in what we know.
We have $\overline{X} = 18$ , $\mu = 20$ (from our null hypothesis), $\sigma = 4.6$ , and $n = 48$ .

STEP 6

Substituting the values into our formula, we get: $z = \frac{18 - 20}{\frac{4.6}{\sqrt{48}}}$ .

STEP 7

Let's simplify the denominator first: $\frac{4.6}{\sqrt{48}} \approx \frac{4.6}{6.928} \approx 0.664$ .
So, our standard error is approximately 0.664.

STEP 8

Now, let's compute the numerator: $18 - 20 = -2$ .
This tells us our sample mean is 2 units below our hypothesized mean.

STEP 9

Finally, divide the numerator by the denominator: $z = \frac{-2}{0.664} \approx -3.01$ .
So, our test statistic is approximately $-3.01$ .

STEP 10

Our test statistic is approximately $-3.01$ .
This means our sample mean is about 3 standard deviations below the hypothesized population mean.

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QuestionH0:μ=20H_0: \mu = 20H0​:μ=20 Ha:μ<20H_a: \mu < 20Ha​:μ<20 Use the following information: n=48n = 48n=48, X‾=18\overline{X} = 18X=18, and σ=4.6\sigma = 4.6σ=4.6 To find the test statistic (Step 2).

Studdy Solution

STEP 1

STEP 2

1. Set up the z-test2. Calculate the test statistic

STEP 3

Alright, so we're dealing with a hypothesis test here!Our **null hypothesis** H0H_0H0​ says the *population mean* μ\muμ is 20.But we suspect it might actually be *less* than 20, and that's our **alternative hypothesis** HaH_aHa​.

STEP 4

STEP 5

Let's plug in what we know.We have X‾=18\overline{X} = 18X=18, μ=20\mu = 20μ=20 (from our null hypothesis), σ=4.6\sigma = 4.6σ=4.6, and n=48n = 48n=48.

STEP 6

Substituting the values into our formula, we get: z=18−204.648z = \frac{18 - 20}{\frac{4.6}{\sqrt{48}}}z=48​4.6​18−20​.

STEP 7

Let's simplify the denominator first: 4.648≈4.66.928≈0.664\frac{4.6}{\sqrt{48}} \approx \frac{4.6}{6.928} \approx 0.66448​4.6​≈6.9284.6​≈0.664.So, our **standard error** is approximately 0.664.

STEP 8

Now, let's compute the numerator: 18−20=−218 - 20 = -218−20=−2.This tells us our **sample mean** is 2 units *below* our **hypothesized mean**.

STEP 9

Finally, divide the numerator by the denominator: z=−20.664≈−3.01z = \frac{-2}{0.664} \approx -3.01z=0.664−2​≈−3.01.So, our **test statistic** is approximately −3.01-3.01−3.01.

STEP 10

Our **test statistic** is approximately −3.01-3.01−3.01.This means our sample mean is about 3 standard deviations *below* the hypothesized population mean.

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Question $H_0: \mu = 20$ $H_a: \mu < 20$ Use the following information: $n = 48$ , $\overline{X} = 18$ , and $\sigma = 4.6$ To find the test statistic (Step 2).

1. Set up the z-test
2. Calculate the test statistic

Alright, so we're dealing with a hypothesis test here!
Our null hypothesis $H_0$ says the population mean $\mu$ is 20.
But we suspect it might actually be less than 20, and that's our alternative hypothesis $H_a$ .

Let's plug in what we know.
We have $\overline{X} = 18$ , $\mu = 20$ (from our null hypothesis), $\sigma = 4.6$ , and $n = 48$ .

Substituting the values into our formula, we get: $z = \frac{18 - 20}{\frac{4.6}{\sqrt{48}}}$ .

Let's simplify the denominator first: $\frac{4.6}{\sqrt{48}} \approx \frac{4.6}{6.928} \approx 0.664$ .
So, our standard error is approximately 0.664.

Now, let's compute the numerator: $18 - 20 = -2$ .
This tells us our sample mean is 2 units below our hypothesized mean.

Finally, divide the numerator by the denominator: $z = \frac{-2}{0.664} \approx -3.01$ .
So, our test statistic is approximately $-3.01$ .

Our test statistic is approximately $-3.01$ .
This means our sample mean is about 3 standard deviations below the hypothesized population mean.