To calculate the test statistic and p-value, we follow these steps:

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Step 1: Determine the Null and Alternative Hypotheses

1. Null hypothesis ($H_0$): The population mean is $\mu_0$ (value provided in your hypothesis).
2. Alternative hypothesis ($H_1$): The population mean is different from $\mu_0$ (two-tailed test).

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Step 2: Gather Sample Data

The sample data provided has $n = 60$. Calculate:

1. Sample mean ($\bar{x}$).
2. Sample standard deviation ($s$).

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Step 3: Calculate the Test Statistic

The formula for the $t$-test statistic is:

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

where:

• $\bar{x}$ is the sample mean.
• $s$ is the sample standard deviation.
• $n$ is the sample size.
• $\mu_0$ is the hypothesized population mean.

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Step 4: Calculate the p-value

The p-value is determined based on the $t$-statistic and degrees of freedom ($df = n - 1$).

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Let me compute these values.import numpy as np from scipy.stats import t

Sample data

data = [ 58.2, 41, 28.6, 53.8, 84.7, 80, 37.6, 65, 88.5, 54.8, 37.6, 46, 77.7, 58.7, 39.4, 49.5, 57.7, 38.5, 42.6, 61.6, 50.6, 67, 55.8, 41.8, 61.6, 58.7, 71.3, 56.3, 65.5, 64, 58.2, 61.1, 80.9, 91.7, 60.6, 73.1, 71.9, 67.5, 52.2, 59.2, 50.1, 83.7, 70.2, 46, 52.2, 60.6, 104.8, 58.7, 66.5, 24, 72.5, 65, 49.5, 69.1, 50.6, 59.2, 65.5, 88.5, 33.2, 33.2 ]

Hypothesized population mean (mu_0)

mu_0 = 50 # Replace with the actual hypothesized mean if different

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 n = len(data)

Calculate t-statistic

t_statistic = (sample_mean - mu_0) / (sample_std / np.sqrt(n))

Calculate p-value (two-tailed)

df = n - 1 # Degrees of freedom p_value = 2 * t.sf(np.abs(t_statistic), df)

t_statistic, p_value(4.533866132476491, 2.8861736207046044e-05)(4.533866132476491, 2.8861736207046044e-05)successThe results are as follows:

• Test statistic: $t = 4.534$ (accurate to three decimal places)
• p-value: $p = 0.0000$ (accurate to four decimal places)

This indicates a significant result if the significance level ($\alpha$) is greater than the p-value.

Would you like a detailed explanation or assistance with interpreting these results? Here are related questions:

1. How is the sample standard deviation calculated?
2. What are the implications of a low p-value in hypothesis testing?
3. How do degrees of freedom affect the calculation of the p-value?
4. How is a two-tailed test different from a one-tailed test?
5. What conclusions can we draw from these results?

Tip: Always ensure your data meets the assumptions of the test you are conducting, such as normality in the population distribution.