Questions: An oceanographer claims that the mean dive duration of a North Atlantic right whale is 11.3 minutes. A random sample of 36 dive durations has a mean of 12.4 minutes and a standard deviation of 2.1 minutes. At α=0.10 is there enough evidence to reject the oceanographer's claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (a) Identify the claim and state H0 and Ha. H0: μ = 11.3 Ha: μ ≠ 11.3 (Type integers or decimals. Do not round.) The claim is the hypothesis. (b) Use technology to Find the standardized test statistic, t t= (Round to two decimals.) (c) Obtain the P-value P= (Round to three decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. H0 because the P-value greater than α. (e) Interpret the decision in the context of the original claim. There enough evidence at the % level of significance to the claim that the mean dive duration of a North Atlantic right whale is minutes.

An oceanographer claims that the mean dive duration of a North Atlantic right whale is 11.3 minutes. A random sample of 36 dive durations has a mean of 12.4 minutes and a standard deviation of 2.1 minutes. At α=0.10 is there enough evidence to reject the oceanographer's claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (a) Identify the claim and state H0 and Ha. H0: μ = 11.3 Ha: μ ≠ 11.3 (Type integers or decimals. Do not round.) The claim is the hypothesis. (b) Use technology to Find the standardized test statistic, t t= (Round to two decimals.) (c) Obtain the P-value P= (Round to three decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. H0 because the P-value greater than α. (e) Interpret the decision in the context of the original claim. There enough evidence at the % level of significance to the claim that the mean dive duration of a North Atlantic right whale is minutes.
Transcript text: An oceanographer claims that the mean dive duration of a North Atlantic right whale is 11.3 minutes. A random sample of 36 dive durations has a mean of 12.4 minutes and a standard deviation of 2.1 minutes. At $\alpha=0.10$ is there enough evidence to reject the oceanographer's claim? Complete parts (a) through (d) below. Assume the population is normally distributed. (a) Identify the claim and state $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$. \[ H_{0} \] $\square$ $\square$ 11.3 $\mathrm{H}_{\mathrm{a}}$ : $\square$ $\square$ 11.3 (Type integers or decimals. Do not round.) The claim is the $\square$ hypothesis. (b) Use technology to Find the standardized test statistic, t \[ t= \] $\square$ (Round to two decimals.) (c) Obtain the $P$-value \[ P= \] $\square$ (Round to three decimal places as needed.) (d) Decide whether to reject or fail to reject the null hypothesis. $\square$ $\mathrm{H}_{0}$ because the P-value $\square$ greater than $\alpha$. (e) Interpret the decision in the context of the original claim. There $\square$ enough evidence at the $\square$ \% level of significance to $\square$ the claim that the mean dive duration of a North Atlantic right whale is $\square$ minutes.
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Solution

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Solution Steps

Step 1: Identify the Hypotheses

The oceanographer claims that the mean dive duration of a North Atlantic right whale is \( \mu = 11.3 \) minutes. We can set up our hypotheses as follows: \[ H_0: \mu = 11.3 \quad \text{(null hypothesis)} \] \[ H_a: \mu \neq 11.3 \quad \text{(alternative hypothesis)} \] The claim is the null hypothesis.

Step 2: Calculate the Test Statistic

To determine if we can reject the null hypothesis, we calculate the standardized test statistic \( Z_{test} \). The standard error \( SE \) is calculated as: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{2.1}{\sqrt{36}} = 0.35 \] The test statistic is then calculated using the formula: \[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{12.4 - 11.3}{0.35} = 3.1429 \]

Step 3: Calculate the P-value

For a two-tailed test, the P-value is calculated as: \[ P = 2 \times (1 - T(|z|)) = 0.0017 \]

Step 4: Decision on the Null Hypothesis

We compare the P-value to the significance level \( \alpha = 0.10 \): Since \( P = 0.0017 < \alpha \), we reject the null hypothesis \( H_0 \).

Final Answer

There is enough evidence at the \( 10\% \) level of significance to reject the claim that the mean dive duration of a North Atlantic right whale is \( 11.3 \) minutes.

\(\boxed{\text{Reject } H_0}\)

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