To test the claim that the times are from a population with a mean equal to 60 seconds, we'll conduct a one-sample t-test. Here's the step-by-step solution:

Part 1: Hypotheses

The null and alternative hypotheses are:

• Null Hypothesis ($H_0$): $\mu = 60$ (The mean time is equal to 60 seconds.)
• Alternative Hypothesis ($H_1$): $\mu \neq 60$ (The mean time is not equal to 60 seconds.)

Part 2: Test Statistic

First, we need to calculate the sample mean ($\bar{x}$) and the sample standard deviation ($s$).

Given the dataTo test the claim that the times are from a population with a mean equal to 60 seconds, we'll conduct a one-sample t-test. Here's the step-by-step solution:

Part 1: Hypotheses

The null and alternative hypotheses are:

• Null Hypothesis ($H_0$): $\mu = 60$ (The mean time is equal to 60 seconds.)
• Alternative Hypothesis ($H_1$): $\mu \neq 60$ (The mean time is not equal to 60 seconds.)

Part 2: Test Statistic

First, we need to calculate the sample mean ($\bar{x}$) and the sample standard deviation ($s$).

Given the $70, 83, 39, 66, 41, 21, 60, 62, 65, 47, 66, 69, 95, 91, 67$

Step 1: Calculate the sample mean ($\bar{x}$)

$\bar{x} = \frac{\sum x_i}{n}$ Where:

• $n = 15$ (the number of observations)
• $\sum x_i$ is the sum of all observations.

Step 2: Calculate the sample standard deviation ($s$)

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Step 3: Calculate the test statistic (t)

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ Where:

• $\mu_0 = 60$ (the population mean under the null hypothesis)

I'll now calculate these values.### Results:

• Sample Mean ($\bar{x}$): 62.8 seconds
• Sample Standard Deviation ($s$): 19.68 seconds (rounded to two decimal places)
• Test Statistic (t): 0.55 (rounded to two decimal places)

Conclusion:

Using the calculated test statistic of $t = 0.55$, you can now compare this with the critical value from the t-distribution table for a two-tailed test at the 0.10 significance level with $n-1 = 14$ degrees of freedom, or directly find the p-value associated with this t-statistic.

Do you want to proceed with finding the p-value or critical value to finalize the hypothesis test? Let me know if you need further details!

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Here are five related questions you might want to explore:

1. How do you determine the degrees of freedom in a t-test?
2. What is the p-value, and how do you interpret it?
3. How does the significance level affect the outcome of a hypothesis test?
4. What assumptions must be met for the t-test to be valid?
5. How would the test results change if the sample size were larger?

Tip: When conducting a hypothesis test, always ensure the assumptions (e.g., normality, independence) are reasonably met to validate your results.