Question

an oceanographer claims that the mean dive depth of a north atlantic right whale is 94 meters. you suspect that this claim is not true. in order to refute the oceanographers claim, you obtain a random sample of 40 dive depths with a mean of 106 meters and a standard deviation of 31.3 meters. state the null and alternative hypotheses of this test. round answers to 4 decimal places. use a significance level of \\( \alpha = 0.01 \\).

a. \\( h _ { 0 } : \\) meters
b. \\( h _ { 1 } : \\) meters
c. the test statistic is
d. the p - value for this hypothesis test is
e. the correct decision is:
\\( \bigcirc \\) reject the null hypothesis.
\\( \bigcirc \\) reject the alternative hypothesis.
\\( \bigcirc \\) do not reject the null hypothesis.
\\( \bigcirc \\) reject the claim.
f. is the result considered significant?
\\( \bigcirc \\) no. the result is not significant because the p - value is \\( \leq \alpha \\).
\\( \bigcirc \\) no. the result is not significant because the p - value is \\( > \alpha \\).
\\( \bigcirc \\) yes. the result is significant because the p - value is \\( \leq \alpha \\).
\\( \bigcirc \\) yes. the result is significant because the p - value is \\( > \alpha \\).
g. we conclude that:
\\( \bigcirc \\) we can dispute the oceanographers claim that the mean dive depth of a north atlantic right whale is 94 meters at the 0.01 level of significance.
\\( \bigcirc \\) we cannot dispute the oceanographers claim that the mean dive depth of a north atlantic right whale is 94 meters at the 0.01 level of significance.

Explanation:

Step1: State null and alternative hypotheses

The null hypothesis \(H_0\) is the claim we assume to be true initially. The oceanographer claims \(\mu = 94\), so \(H_0:\mu=94.0000\) meters. The alternative hypothesis \(H_1\) is what we suspect. Since we suspect the claim is not true (a two - tailed test), \(H_1:\mu
eq94.0000\) meters.

Step2: Calculate the test statistic

The formula for the test statistic \(t\) in a one - sample \(t\) - test (since population standard deviation is unknown) is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\). Here, \(\bar{x} = 106\), \(\mu = 94\), \(s = 31.3\), \(n = 40\).

$$t=\frac{106 - 94}{31.3/\sqrt{40}}$$

$$t=\frac{12}{31.3/6.3246}$$

$$t=\frac{12}{4.9489}$$

\(t = 2.4247\)

Step3: Calculate the p - value

For a two - tailed \(t\) - test with \(n-1=40 - 1=39\) degrees of freedom. Using a \(t\) - distribution table or software (e.g., in R: \(2*(1 - pt(2.4247,39))\)), the p - value is \(0.0196\)

Step4: Make a decision

Since \(\alpha=0.01\) and \(p - value=0.0196>0.01\), we do not reject the null hypothesis.

Step5: Determine significance

The result is not significant because the \(p - value> \alpha\)

Step6: Conclusion

We cannot dispute the oceanographer’s claim that the mean dive depth of a North Atlantic right whale is 94 meters at the 0.01 level of significance.

Answer:

a. \(H_0:\mu = 94.0000\) meters
b. \(H_1:\mu
eq94.0000\) meters
c. The test statistic is \(2.4247\)
d. The p - value for this hypothesis test is \(0.0196\)
e. Do not reject the null hypothesis
f. No. The result is not significant because the \(p - value>\alpha\)
g. We cannot dispute the oceanographer’s claim that the mean dive depth of a North Atlantic right whale is 94 meters at the 0.01 level of significance.