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It seems like you're describing a scenario where you have test scores with a known mean and standard deviation, and you want to perform a hypothesis test. Here's how we can break it down:\r
\n" ); document.write( "\n" ); document.write( "**Understanding the Information**\r
\n" ); document.write( "\n" ); document.write( "* **Mean (μ):** The average test score is 65.
\n" ); document.write( "* **Standard Deviation (σ):** The spread of the scores is 5. This tells us how much the scores typically deviate from the mean.
\n" ); document.write( "* **Possible Score Range:** Students can score between 0 and 100 on the test.
\n" ); document.write( "* **Alpha (α):** This is the significance level, set at 0.01. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error).
\n" ); document.write( "* **Two-tailed:** This means we're interested in deviations both above and below the mean.\r
\n" ); document.write( "\n" ); document.write( "**Setting Up the Hypothesis Test**\r
\n" ); document.write( "\n" ); document.write( "Since we don't have a specific claim to test, let's assume we want to test whether the true population mean of the test scores is different from a certain value. Let's say we want to test if the population mean is different from 70.\r
\n" ); document.write( "\n" ); document.write( "1. **Null Hypothesis (H0):** The population mean is equal to 70 (μ = 70).
\n" ); document.write( "2. **Alternative Hypothesis (H1):** The population mean is not equal to 70 (μ ≠ 70).\r
\n" ); document.write( "\n" ); document.write( "**Steps to Perform the Test**\r
\n" ); document.write( "\n" ); document.write( "Since we know the population standard deviation, we can use a z-test.\r
\n" ); document.write( "\n" ); document.write( "1. **Calculate the z-score:** z = (x̄ - μ) / (σ / √n) \r
\n" ); document.write( "\n" ); document.write( " Where:
\n" ); document.write( " * x̄ is the sample mean (we would need a sample mean to actually perform the test)
\n" ); document.write( " * μ is the hypothesized population mean (70 in this case)
\n" ); document.write( " * σ is the population standard deviation (5)
\n" ); document.write( " * n is the sample size (we would need a sample size to perform the test)\r
\n" ); document.write( "\n" ); document.write( "2. **Find the critical z-values:** Since it's a two-tailed test with α = 0.01, we need to find the z-values that cut off 0.005 (0.01/2) in each tail of the standard normal distribution. Using a z-table or calculator, the critical z-values are approximately ±2.576.\r
\n" ); document.write( "\n" ); document.write( "3. **Compare the calculated z-score to the critical z-values:**
\n" ); document.write( " * If the calculated z-score falls outside the range of -2.576 to 2.576, we reject the null hypothesis.
\n" ); document.write( " * If the calculated z-score falls within the range of -2.576 to 2.576, we fail to reject the null hypothesis.\r
\n" ); document.write( "\n" ); document.write( "**Additional Considerations**\r
\n" ); document.write( "\n" ); document.write( "* **Sample Data:** To actually perform the hypothesis test, you would need a sample of test scores.
\n" ); document.write( "* **Effect Size:** You could also calculate Cohen's d to measure the effect size, which indicates the standardized difference between the sample mean and the hypothesized population mean.\r
\n" ); document.write( "\n" ); document.write( "**Let me know if you have a specific sample of test scores or a different value for the hypothesized population mean, and I can help you perform the complete hypothesis test.** \n" ); document.write( "