Solved on Nov 16, 2023

Test if more than 91% of acid reflux patients are healed in 8 weeks using drug. $\alpha=0.01$ . Find test statistic $z_0=0.72$ and P-value.

STEP 1

Assumptions1. The manufacturer claims that more than91% of patients are healed within8 weeks. . In clinical trials,217 out of235 patients were healed after8 weeks.
3. We are testing the manufacturer's claim at the $\alpha=0.01$ level of significance.
4. We are asked to find the test statistic, $z_{0}$ , and the-value.

STEP 2

First, we need to calculate the sample proportion ( $\hat{p}$ ), which is the proportion of patients healed in the clinical trials. This is calculated by dividing the number of successful outcomes (patients healed) by the total number of trials (total patients).
$\hat{p} = \frac{Number\, of\, successful\, outcomes}{Total\, number\, of\, trials}$

STEP 3

Now, plug in the given values for the number of successful outcomes and total number of trials to calculate the sample proportion.
$\hat{p} = \frac{217}{235}$

STEP 4

Calculate the sample proportion.
$\hat{p} = \frac{217}{235} \approx0.9234$

STEP 5

Next, we need to calculate the population proportion ( $p$ ), which is the proportion claimed by the manufacturer.
$p =0.91$

STEP 6

Now, we calculate the standard error ( $SE$ ) using the formula $SE = \sqrt{\frac{p(1-p)}{n}}$ where $n$ is the total number of trials.

STEP 7

Plug in the values for $p$ and $n$ to calculate the standard error.
$SE = \sqrt{\frac{0.91(1-0.91)}{235}}$

STEP 8

Calculate the standard error.
$SE \approx0.0178$

STEP 9

Now, we calculate the test statistic $z_{}$ using the formula $z_{} = \frac{\hat{p} - p}{SE}$

STEP 10

Plug in the values for $\hat{p}$ , $p$ , and $SE$ to calculate the test statistic.
$z_{0} = \frac{0.9234 -0.91}{0.0178}$

STEP 11

Calculate the test statistic.
$z_{0} \approx0.72$

STEP 12

Now, we calculate the-value. The-value is the probability that we would observe a result as extreme as the test statistic, assuming the null hypothesis is true. Since the manufacturer's claim is that more than91% of patients are healed, this is a one-tailed test. We use the standard normal distribution (Z-distribution) to find the-value.
$-value =(Z > z_{0})$

STEP 13

Plug in the value for $z_{0}$ to calculate the-value.
$-value =(Z >0.72)$

STEP 14

Calculate the-value. $-value \approx0.236$

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Test if more than 91% of acid reflux patients are healed in 8 weeks using drug. $\alpha=0.01$ . Find test statistic $z_0=0.72$ and P-value.

STEP 1

STEP 2

STEP 3

Now, plug in the given values for the number of successful outcomes and total number of trials to calculate the sample proportion.
$\hat{p} = \frac{217}{235}$

STEP 4

Calculate the sample proportion.
$\hat{p} = \frac{217}{235} \approx0.9234$

STEP 5

Next, we need to calculate the population proportion ( $p$ ), which is the proportion claimed by the manufacturer.
$p =0.91$

STEP 6

Now, we calculate the standard error ( $SE$ ) using the formula $SE = \sqrt{\frac{p(1-p)}{n}}$ where $n$ is the total number of trials.

STEP 7

Plug in the values for $p$ and $n$ to calculate the standard error.
$SE = \sqrt{\frac{0.91(1-0.91)}{235}}$

STEP 8

Calculate the standard error.
$SE \approx0.0178$

STEP 9

Now, we calculate the test statistic $z_{}$ using the formula $z_{} = \frac{\hat{p} - p}{SE}$

STEP 10

Plug in the values for $\hat{p}$ , $p$ , and $SE$ to calculate the test statistic.
$z_{0} = \frac{0.9234 -0.91}{0.0178}$

STEP 11

Calculate the test statistic.
$z_{0} \approx0.72$

STEP 12

STEP 13

Plug in the value for $z_{0}$ to calculate the-value.
$-value =(Z >0.72)$

STEP 14

Calculate the-value. $-value \approx0.236$

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Test if more than 91% of acid reflux patients are healed in 8 weeks using drug. α=0.01\alpha=0.01α=0.01. Find test statistic z0=0.72z_0=0.72z0​=0.72 and P-value.

STEP 1

STEP 2

STEP 3

Now, plug in the given values for the number of successful outcomes and total number of trials to calculate the sample proportion.p^=217235\hat{p} = \frac{217}{235}p^​=235217​

STEP 4

Calculate the sample proportion.p^=217235≈0.9234\hat{p} = \frac{217}{235} \approx0.9234p^​=235217​≈0.9234

STEP 5

Next, we need to calculate the population proportion (ppp), which is the proportion claimed by the manufacturer.p=0.91p =0.91p=0.91

STEP 6

Now, we calculate the standard error (SESESE) using the formulaSE=p(1−p)nSE = \sqrt{\frac{p(1-p)}{n}}SE=np(1−p)​​where nnn is the total number of trials.

STEP 7

Plug in the values for ppp and nnn to calculate the standard error.SE=0.91(1−0.91)235SE = \sqrt{\frac{0.91(1-0.91)}{235}}SE=2350.91(1−0.91)​​

STEP 8

Calculate the standard error.SE≈0.0178SE \approx0.0178SE≈0.0178

STEP 9

Now, we calculate the test statistic zz_{}z​ using the formulaz=p^−pSEz_{} = \frac{\hat{p} - p}{SE}z​=SEp^​−p​

STEP 10

Plug in the values for p^\hat{p}p^​, ppp, and SESESE to calculate the test statistic.z0=0.9234−0.910.0178z_{0} = \frac{0.9234 -0.91}{0.0178}z0​=0.01780.9234−0.91​

STEP 11

Calculate the test statistic.z0≈0.72z_{0} \approx0.72z0​≈0.72

STEP 12

STEP 13

Plug in the value for z0z_{0}z0​ to calculate the-value.−value=(Z>0.72)-value =(Z >0.72)−value=(Z>0.72)

STEP 14

Calculate the-value.−value≈0.236-value \approx0.236−value≈0.236

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Test if more than 91% of acid reflux patients are healed in 8 weeks using drug. $\alpha=0.01$ . Find test statistic $z_0=0.72$ and P-value.

Now, plug in the given values for the number of successful outcomes and total number of trials to calculate the sample proportion.
$\hat{p} = \frac{217}{235}$

Calculate the sample proportion.
$\hat{p} = \frac{217}{235} \approx0.9234$

Next, we need to calculate the population proportion ( $p$ ), which is the proportion claimed by the manufacturer.
$p =0.91$

Now, we calculate the standard error ( $SE$ ) using the formula $SE = \sqrt{\frac{p(1-p)}{n}}$ where $n$ is the total number of trials.

Plug in the values for $p$ and $n$ to calculate the standard error.
$SE = \sqrt{\frac{0.91(1-0.91)}{235}}$

Calculate the standard error.
$SE \approx0.0178$

Now, we calculate the test statistic $z_{}$ using the formula $z_{} = \frac{\hat{p} - p}{SE}$

Plug in the values for $\hat{p}$ , $p$ , and $SE$ to calculate the test statistic.
$z_{0} = \frac{0.9234 -0.91}{0.0178}$

Calculate the test statistic.
$z_{0} \approx0.72$

Plug in the value for $z_{0}$ to calculate the-value.
$-value =(Z >0.72)$

Calculate the-value. $-value \approx0.236$