Math / Data & Statistics

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PROBCEM 6 Weatherwise is a magazine published by the American Meteorological Society. One issue gives a rating system used lo classify Nor'easter storms that frequently hit New England and can cause much damage near the ocean. A severe storm has an average peak wave height of $\mu=16.4$ feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. Peak wave heights are usually measured from land (using binoculars) off fixed cement piers. Suppose that a reading of 36 waves showed an average wave height of $\bar{x}=17.3$ feets. Previous studies of severe storms indicate that $\sigma=3.5$ feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating?
Use the 0.01 level of significance and the $\mathbf{P}$ - Value method a.) Hypotheses (5 points): $\begin{array}{l} H_{0} \cdot \mu=16.4 \text { feet } \\ H_{a} \cdot \mu>16.4 \text { feet } \end{array}$
Levelot S Sandicance: $\alpha=0.01$ Sample sie $n=36$ $z=\frac{0.9}{3.5 / 6}=\frac{0.9}{0.5833} \approx 1.51 \sqrt{n}$ $P(z>1.54) \approx 0.0618$ d.) Decision ( 5 points):
Since the $P$ -value 0.0618 is greater than the sio level $a=0.01$ , we fail to reject the null hypo e.) Conclusion (5 points)

Studdy Solution

STEP 1

1. The null hypothesis $H_0: \mu = 16.4$ feet represents the average peak wave height for a severe storm.
2. The alternative hypothesis $H_a: \mu > 16.4$ feet suggests the storm is increasing in severity.
3. The sample mean wave height is $\bar{x} = 17.3$ feet.
4. The population standard deviation is $\sigma = 3.5$ feet.
5. The sample size is $n = 36$ .
6. The significance level is $\alpha = 0.01$ .

STEP 2

1. State the hypotheses.
2. Calculate the test statistic.
3. Determine the $P$ -value.
4. Make a decision based on the $P$ -value.
5. Draw a conclusion.

STEP 3

State the null and alternative hypotheses:
\begin{align} H_0: & \quad \mu = 16.4 \text{ feet} \\ H_a: & \quad \mu > 16.4 \text{ feet} \end{align}

STEP 4

Calculate the test statistic using the formula for the $z$ -score:
$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$
Substitute the given values:
$z = \frac{17.3 - 16.4}{3.5 / \sqrt{36}} = \frac{0.9}{3.5 / 6} = \frac{0.9}{0.5833} \approx 1.54$

STEP 5

Determine the $P$ -value for $z > 1.54$ :
Using standard normal distribution tables or a calculator, find:
$P(z > 1.54) \approx 0.0618$

STEP 6

Make a decision based on the $P$ -value:
Since the $P$ -value $0.0618$ is greater than the significance level $\alpha = 0.01$ , we fail to reject the null hypothesis.

STEP 7

Draw a conclusion:
There is not enough statistical evidence at the $\alpha = 0.01$ level to suggest that the storm is increasing above the severe rating.

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Studdy Solution

STEP 1

STEP 2

1. State the hypotheses.
2. Calculate the test statistic.
3. Determine the $P$ -value.
4. Make a decision based on the $P$ -value.
5. Draw a conclusion.

STEP 3

State the null and alternative hypotheses:
\begin{align} H_0: & \quad \mu = 16.4 \text{ feet} \\ H_a: & \quad \mu > 16.4 \text{ feet} \end{align}

STEP 4

Calculate the test statistic using the formula for the $z$ -score:
$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$
Substitute the given values:
$z = \frac{17.3 - 16.4}{3.5 / \sqrt{36}} = \frac{0.9}{3.5 / 6} = \frac{0.9}{0.5833} \approx 1.54$

STEP 5

Determine the $P$ -value for $z > 1.54$ :
Using standard normal distribution tables or a calculator, find:
$P(z > 1.54) \approx 0.0618$

STEP 6

Make a decision based on the $P$ -value:
Since the $P$ -value $0.0618$ is greater than the significance level $\alpha = 0.01$ , we fail to reject the null hypothesis.

STEP 7

Draw a conclusion:
There is not enough statistical evidence at the $\alpha = 0.01$ level to suggest that the storm is increasing above the severe rating.

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Studdy Solution

STEP 1

STEP 2

1. State the hypotheses.2. Calculate the test statistic.3. Determine the P P P-value.4. Make a decision based on the P P P-value.5. Draw a conclusion.

STEP 3

State the null and alternative hypotheses:\begin{align*} H_0: & \quad \mu = 16.4 \text{ feet} \\ H_a: & \quad \mu > 16.4 \text{ feet} \end{align*}

STEP 4

STEP 5

Determine the P P P-value for z>1.54 z > 1.54 z>1.54:Using standard normal distribution tables or a calculator, find:P(z>1.54)≈0.0618P(z > 1.54) \approx 0.0618P(z>1.54)≈0.0618

STEP 6

Make a decision based on the P P P-value:Since the P P P-value 0.0618 0.0618 0.0618 is greater than the significance level α=0.01 \alpha = 0.01 α=0.01, we fail to reject the null hypothesis.

STEP 7

Draw a conclusion:There is not enough statistical evidence at the α=0.01 \alpha = 0.01 α=0.01 level to suggest that the storm is increasing above the severe rating.

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1. State the hypotheses.
2. Calculate the test statistic.
3. Determine the $P$ -value.
4. Make a decision based on the $P$ -value.
5. Draw a conclusion.

State the null and alternative hypotheses:
\begin{align} H_0: & \quad \mu = 16.4 \text{ feet} \\ H_a: & \quad \mu > 16.4 \text{ feet} \end{align}

Determine the $P$ -value for $z > 1.54$ :
Using standard normal distribution tables or a calculator, find:
$P(z > 1.54) \approx 0.0618$

Make a decision based on the $P$ -value:
Since the $P$ -value $0.0618$ is greater than the significance level $\alpha = 0.01$ , we fail to reject the null hypothesis.

Draw a conclusion:
There is not enough statistical evidence at the $\alpha = 0.01$ level to suggest that the storm is increasing above the severe rating.