Math / Data & Statistics

QuestionA drug, which is used for treating cancer, has potentially dangerous side effects if it is taken in doses which are larger than the required dosage for the treatment. The tablet should contain 54.94 mg and the standard deviation should be 0.04 . 25 tablets are randomly selected and the amount of drug in each tablet is measured. The sample has a mean of 54.945 mg and a variance of 0.0004 mg . Does the data suggest at $\alpha=0.025$ that the tablets vary by less than the desired amount?
Step 2 of 5 : Determine the critical value(s) of the test statistic. If the test is two-tailed, separate the values with a comma. Round your answer to three decimal places.
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Studdy Solution

STEP 1

What is this asking? Are these cancer drug tablets really as precise as we need them to be, or are they varying too much in their dosage? Watch out! Don't mix up variance and standard deviation!
Also, remember that we're looking at a sample of tablets, not the entire population.

STEP 2

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

STEP 3

We're dealing with a variance here, so we'll use a chi-squared test.
Since we want to see if the variance is less than the desired amount, this is a one-tailed test to the left.

STEP 4

Our null hypothesis ( $H_0$ ) is that the population variance ( $\sigma^2$ ) is equal to the desired variance.
The desired standard deviation is $0.04$ mg, so the desired variance is $(0.04)^2 = 0.0016$ mg $^2$ .
So, $H_0: \sigma^2 = 0.0016$ .

STEP 5

Our alternative hypothesis ( $H_1$ ) is that the population variance is less than $0.0016$ mg $^2$ .
So, $H_1: \sigma^2 < 0.0016$ .

STEP 6

Our significance level ( $\alpha$ ) is $\alpha = 0.025$ .
This means we're willing to accept a 2.5% chance of rejecting the null hypothesis when it's actually true.

STEP 7

Since this is a left-tailed test with $\alpha = 0.025$ , we need to find the chi-squared value that corresponds to an area of $0.025$ in the left tail.

STEP 8

Our degrees of freedom are the sample size minus 1.
We have $25$ tablets, so our degrees of freedom are $25 - 1 = 24$ .

STEP 9

Looking up the chi-squared distribution table with $24$ degrees of freedom and a left-tail area of $0.025$ , we find our critical value to be approximately $12.401$ .

STEP 10

The formula for the chi-squared test statistic is $\chi^2 = \frac{(n-1) \cdot s^2}{\sigma^2}$ where $n$ is the sample size, $s^2$ is the sample variance, and $\sigma^2$ is the population variance under the null hypothesis.

STEP 11

We know $n = 25$ , $s^2 = 0.0004$ , and $\sigma^2 = 0.0016$ .
Plugging these values into the formula, we get: $\chi^2 = \frac{(25-1) \cdot 0.0004}{0.0016} = \frac{24 \cdot 0.0004}{0.0016} = \frac{0.0096}{0.0016} = 6$ So, our test statistic is $6$ .

STEP 12

Our test statistic ( $6$ ) is less than our critical value ( $12.401$ ).

STEP 13

Since our test statistic falls in the rejection region (to the left of the critical value), we reject the null hypothesis.

STEP 14

The data suggests at $\alpha = 0.025$ that the tablets vary by less than the desired amount.
The calculated test statistic of $6$ falls within the rejection region defined by the critical value of $12.401$ .

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Studdy Solution

STEP 1

What is this asking? Are these cancer drug tablets really as precise as we need them to be, or are they varying too much in their dosage? Watch out! Don't mix up variance and standard deviation!
Also, remember that we're looking at a sample of tablets, not the entire population.

STEP 2

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

STEP 3

We're dealing with a variance here, so we'll use a chi-squared test.
Since we want to see if the variance is less than the desired amount, this is a one-tailed test to the left.

STEP 4

Our null hypothesis ( $H_0$ ) is that the population variance ( $\sigma^2$ ) is equal to the desired variance.
The desired standard deviation is $0.04$ mg, so the desired variance is $(0.04)^2 = 0.0016$ mg $^2$ .
So, $H_0: \sigma^2 = 0.0016$ .

STEP 5

Our alternative hypothesis ( $H_1$ ) is that the population variance is less than $0.0016$ mg $^2$ .
So, $H_1: \sigma^2 < 0.0016$ .

STEP 6

Our significance level ( $\alpha$ ) is $\alpha = 0.025$ .
This means we're willing to accept a 2.5% chance of rejecting the null hypothesis when it's actually true.

STEP 7

Since this is a left-tailed test with $\alpha = 0.025$ , we need to find the chi-squared value that corresponds to an area of $0.025$ in the left tail.

STEP 8

Our degrees of freedom are the sample size minus 1.
We have $25$ tablets, so our degrees of freedom are $25 - 1 = 24$ .

STEP 9

Looking up the chi-squared distribution table with $24$ degrees of freedom and a left-tail area of $0.025$ , we find our critical value to be approximately $12.401$ .

STEP 10

The formula for the chi-squared test statistic is $\chi^2 = \frac{(n-1) \cdot s^2}{\sigma^2}$ where $n$ is the sample size, $s^2$ is the sample variance, and $\sigma^2$ is the population variance under the null hypothesis.

STEP 11

We know $n = 25$ , $s^2 = 0.0004$ , and $\sigma^2 = 0.0016$ .
Plugging these values into the formula, we get: $\chi^2 = \frac{(25-1) \cdot 0.0004}{0.0016} = \frac{24 \cdot 0.0004}{0.0016} = \frac{0.0096}{0.0016} = 6$ So, our test statistic is $6$ .

STEP 12

Our test statistic ( $6$ ) is less than our critical value ( $12.401$ ).

STEP 13

Since our test statistic falls in the rejection region (to the left of the critical value), we reject the null hypothesis.

STEP 14

The data suggests at $\alpha = 0.025$ that the tablets vary by less than the desired amount.
The calculated test statistic of $6$ falls within the rejection region defined by the critical value of $12.401$ .

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Studdy Solution

STEP 1

What is this asking? Are these cancer drug tablets *really* as precise as we need them to be, or are they varying too much in their dosage? Watch out! Don't mix up variance and standard deviation!Also, remember that we're looking at a *sample* of tablets, not the entire population.

STEP 2

1. Set up the hypothesis test2. Find the critical value3. Calculate the test statistic4. Make a decision

STEP 3

We're dealing with a variance here, so we'll use a chi-squared test.Since we want to see if the variance is *less than* the desired amount, this is a **one-tailed test** to the left.

STEP 4

STEP 5

Our **alternative hypothesis** (H1H_1H1​) is that the population variance is *less than* 0.00160.00160.0016 mg2^22.So, H1:σ2<0.0016H_1: \sigma^2 < 0.0016H1​:σ2<0.0016.

STEP 6

Our **significance level** (α\alphaα) is α=0.025\alpha = 0.025α=0.025.This means we're willing to accept a 2.5% chance of rejecting the null hypothesis when it's actually true.

STEP 7

Since this is a left-tailed test with α=0.025\alpha = 0.025α=0.025, we need to find the chi-squared value that corresponds to an area of 0.0250.0250.025 in the *left* tail.

STEP 8

Our **degrees of freedom** are the sample size minus 1.We have 252525 tablets, so our degrees of freedom are 25−1=2425 - 1 = 2425−1=24.

STEP 9

Looking up the chi-squared distribution table with 242424 degrees of freedom and a left-tail area of 0.0250.0250.025, we find our **critical value** to be approximately 12.40112.40112.401.

STEP 10

The formula for the chi-squared test statistic is χ2=(n−1)⋅s2σ2\chi^2 = \frac{(n-1) \cdot s^2}{\sigma^2}χ2=σ2(n−1)⋅s2​ where nnn is the sample size, s2s^2s2 is the sample variance, and σ2\sigma^2σ2 is the population variance under the null hypothesis.

STEP 11

STEP 12

Our **test statistic** (666) is less than our **critical value** (12.40112.40112.401).

STEP 13

Since our test statistic falls in the rejection region (to the left of the critical value), we **reject the null hypothesis**.

STEP 14

The data suggests at α=0.025\alpha = 0.025α=0.025 that the tablets vary by *less* than the desired amount.The calculated test statistic of 666 falls within the rejection region defined by the critical value of 12.40112.40112.401.

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What is this asking? Are these cancer drug tablets really as precise as we need them to be, or are they varying too much in their dosage? Watch out! Don't mix up variance and standard deviation!
Also, remember that we're looking at a sample of tablets, not the entire population.

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

We're dealing with a variance here, so we'll use a chi-squared test.
Since we want to see if the variance is less than the desired amount, this is a one-tailed test to the left.

Our alternative hypothesis ( $H_1$ ) is that the population variance is less than $0.0016$ mg $^2$ .
So, $H_1: \sigma^2 < 0.0016$ .

Our significance level ( $\alpha$ ) is $\alpha = 0.025$ .
This means we're willing to accept a 2.5% chance of rejecting the null hypothesis when it's actually true.

Since this is a left-tailed test with $\alpha = 0.025$ , we need to find the chi-squared value that corresponds to an area of $0.025$ in the left tail.

Our degrees of freedom are the sample size minus 1.
We have $25$ tablets, so our degrees of freedom are $25 - 1 = 24$ .

Looking up the chi-squared distribution table with $24$ degrees of freedom and a left-tail area of $0.025$ , we find our critical value to be approximately $12.401$ .

The formula for the chi-squared test statistic is $\chi^2 = \frac{(n-1) \cdot s^2}{\sigma^2}$ where $n$ is the sample size, $s^2$ is the sample variance, and $\sigma^2$ is the population variance under the null hypothesis.

Our test statistic ( $6$ ) is less than our critical value ( $12.401$ ).

Since our test statistic falls in the rejection region (to the left of the critical value), we reject the null hypothesis.

The data suggests at $\alpha = 0.025$ that the tablets vary by less than the desired amount.
The calculated test statistic of $6$ falls within the rejection region defined by the critical value of $12.401$ .