Dive Duration An oceanographer claims that the mean dive duration of a North Atlantic right whale is 11.5 minutes. A random sample of 34 dive durations has a mean of 12.2 minutes and a standard deviation of 2.2 minutes. Is there enough evidence to reject the claim at α=0.10?

Verified step by step guidance

1. Formulate the null hypothesis (H₀) and the alternative hypothesis (Hₐ). H₀: μ = 11.5 (the mean dive duration is 11.5 minutes), Hₐ: μ ≠ 11.5 (the mean dive duration is not 11.5 minutes). This is a two-tailed test.
2. Determine the test statistic to use. Since the population standard deviation is unknown and the sample size is greater than 30, use the t-test. The formula for the t-test statistic is: t = (x̄ - μ₀) / (s / √n), where x̄ is the sample mean, μ₀ is the hypothesized mean, s is the sample standard deviation, and n is the sample size.
3. Substitute the given values into the formula. Here, x̄ = 12.2, μ₀ = 11.5, s = 2.2, and n = 34. Compute the standard error (SE) first: SE = s / √n.
4. Determine the critical t-value for a two-tailed test at α = 0.10 with degrees of freedom (df) = n - 1 = 34 - 1 = 33. Use a t-distribution table or statistical software to find the critical t-value.
5. Compare the calculated t-test statistic to the critical t-value. If the absolute value of the t-test statistic is greater than the critical t-value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1). In this case, the null hypothesis would state that the mean dive duration is 11.5 minutes, while the alternative would suggest it is different. The goal is to determine if the sample data provides sufficient evidence to reject the null hypothesis.

P-value

The p-value is a measure that helps determine the significance of the results in hypothesis testing. It represents the probability of obtaining a sample mean at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the significance level (α), in this case, 0.10, it indicates strong evidence against the null hypothesis, leading to its rejection.

Confidence Intervals

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. In this scenario, constructing a confidence interval for the mean dive duration can provide insight into whether the true mean could be 11.5 minutes. If the interval does not include this value, it supports the rejection of the null hypothesis.