Based on the uploaded image, we are working with a hypothesis testing problem. Let's analyze and calculate step by step:

Hypothesis:

• Null Hypothesis ($H_0$): $\mu = 78.1$
• Alternative Hypothesis ($H_a$): $\mu > 78.1$

This is a one-tailed test, and we are testing whether the population mean is greater than 78.1.

Data provided:

Sample data: 83.7, 65.2, 111, 80, 96.7, 75.3

You are asked to calculate:

1. Test statistic (accurate to three decimal places)
2. p-value (accurate to four decimal places)

Step 1: Calculate the sample mean ($\bar{x}$) and standard deviation ($s$):

To calculate these: $\bar{x} = \frac{\text{Sum of all sample values}}{\text{Number of values}}$ $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Step 2: Compute the t-test statistic:

The formula for the t-test statistic is: $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$ Where:

• $\bar{x}$ = sample mean
• $\mu = 78.1$ = hypothesized mean
• $s$ = sample standard deviation
• $n = 6$ = sample size

Step 3: Find the p-value:

Using the t-test statistic and degrees of freedom ($df = n-1 = 5$), calculate the one-tailed p-value.

Let me calculate this for you.import numpy as np from scipy.stats import t

Given data

data = np.array([83.7, 65.2, 111, 80, 96.7, 75.3]) mu = 78.1 # Hypothesized mean n = len(data) # Sample size

Step 1: Calculate sample mean and standard deviation

sample_mean = np.mean(data) sample_std = np.std(data, ddof=1) # Use ddof=1 for sample standard deviation

Step 2: Calculate the t-statistic

t_statistic = (sample_mean - mu) / (sample_std / np.sqrt(n))

Step 3: Calculate the p-value (one-tailed)

df = n - 1 # Degrees of freedom p_value = 1 - t.cdf(t_statistic, df)

(sample_mean, sample_std, t_statistic, p_value)(85.31666666666666, 16.279486068874125, 1.0858543631106046, 0.1635493256750673)(85.31666666666666, 16.279486068874125, 1.0858543631106046, 0.1635493256750673)successHere are the calculated results:

1. Sample Mean ($\bar{x}$): $85.317$ (rounded to three decimal places)
2. Sample Standard Deviation ($s$): $16.279$ (rounded to three decimal places)
3. Test Statistic ($t$): $1.086$ (rounded to three decimal places)
4. p-value: $0.1635$ (rounded to four decimal places)

Interpretation:

• The p-value is $0.1635$, which is greater than the commonly used significance level ($\alpha = 0.05$).
• Thus, we fail to reject the null hypothesis. This means there is not enough evidence to conclude that the population mean is greater than $78.1$.

Would you like a further breakdown or visualization of the calculations?

Related Questions:

1. How is the t-distribution used in hypothesis testing?
2. What happens to the p-value if the sample size increases?
3. How does the standard deviation affect the t-test result?
4. Can this test be performed using a z-test instead? Why or why not?
5. What are the assumptions of the one-sample t-test?

Tip: Always confirm that your data meets the assumptions of normality when using a t-test for small sample sizes.