Question

an oceanographer claims that the mean dive duration of a north atlantic right whale is 11.6 minutes. a random sample of 35 dive durations has a mean of 12.3 minutes and a standard deviation of 2.2 minutes. at \\( \alpha = 0.01 \\) is there enough evidence to reject the oceanographers claim? complete parts (a) through (d) below. assume the population is normally distributed.
(a) identify the claim and state \\( h_0 \\) and \\( h_a \\).
\\( h_0: \mu = 11.6 \\)
\\( h_a: \mu \
eq 11.6 \\)
(type integers or decimals. do not round.)
the claim is the null hypothesis.
(b) use technology to find the p - value. find the standardized test statistic, t.
\\( t = \square \\)
(round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for the t - statistic

The formula for the t - statistic in a one - sample t - test is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean under the null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step2: Identify the values from the problem

We are given that \(\bar{x} = 12.3\), \(\mu=11.6\), \(s = 2.2\), and \(n = 35\).

Step3: Substitute the values into the formula

$$
LATEXBLOCK0
$$

First, calculate \(\sqrt{35}\approx5.916\), then \(2.2/\sqrt{35}\approx2.2\div5.916\approx0.372\)

$$ t=\frac{0.7}{0.372}\approx1.88 $$

Answer:

\(t = 1.88\)