To determine whether the dissolved oxygen content is significantly less than 5 ppm, we conduct a one-sample $t$-test for the mean. Here are the steps:

Step 1: State the Hypotheses

• Null hypothesis ($H_0$): $\mu = 5$ ppm (mean dissolved oxygen content is 5 ppm).
• Alternative hypothesis ($H_a$): $\mu < 5$ ppm (mean dissolved oxygen content is less than 5 ppm).

Step 2: Calculate the Test Statistic

The formula for the test statistic is:

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

where:

• $\bar{x}$: Sample mean
• $\mu_0$: Population mean under the null hypothesis (5 ppm)
• $s$: Sample standard deviation
• $n$: Sample size

Step 3: Calculate Sample Mean ($\bar{x}$) and Standard Deviation ($s$)

The data are: $5.0, 5.2, 4.8, 4.9, 5.1, 4.8$.

1. Compute the mean: $\bar{x} = \frac{\sum x_i}{n} = \frac{5.0 + 5.2 + 4.8 + 4.9 + 5.1 + 4.8}{6}$

2. Compute the standard deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Let me calculate these values.import numpy as np

Data

data = np.array([5.0, 5.2, 4.8, 4.9, 5.1, 4.8])

Calculations

mean = np.mean(data) std_dev = np.std(data, ddof=1) # Sample standard deviation n = len(data)

mean, std_dev(4.966666666666667, 0.16329931618554525)(4.966666666666667, 0.16329931618554525)successThe sample mean ($\bar{x}$) is approximately $4.967$, and the sample standard deviation ($s$) is approximately $0.163$.

Step 4: Compute the $t$-Statistic

Now, substitute these values into the formula:

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

Let me compute the $t$-statistic.# Population mean under H0 mu_0 = 5

Calculate the t-statistic

t_statistic = (mean - mu_0) / (std_dev / np.sqrt(n)) t_statistic-0.499999999999998-0.499999999999998successThe $t$-statistic is approximately $-0.50$.

Would you like to proceed with determining the critical value and making a conclusion, or do you need clarification on any steps so far?

Here are some follow-up questions:

1. What is the critical value for a one-tailed test at $\alpha = 0.05$ for this sample size?
2. How is the $t$-distribution used to assess the p-value in hypothesis testing?
3. What assumptions are made in a one-sample $t$-test?
4. How would the test outcome change if the sample size were larger?
5. How does the standard deviation affect the $t$-statistic?

Tip: Always check if the assumptions of normality or sample size adequacy are met when performing $t$-tests!