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An industrial chemist is checking for the desired concentration of a certain chemical component in a batch of polymer blend. The target concentration is set at 8.2%. To verify whether the actual average concentration matches the target for the current production line using a significance level of $\alpha = 0.05$, 15 independently obtained samples are taken for analysis. Assume that the concentration of the chemical component in a sample is normally distributed with $\sigma = 0.5\%$ and that the sample mean is $\bar{x} = 8.0$. 1. Based on the information provided, perform a hypothesis test to determine if there is statistical evidence that the true average concentration of the chemical component is different from 8.2%. State the null and alternative hypotheses and calculate the P-value. Based on the P-value and the significance level, explain your conclusion. (10 points) 2. Assuming that the actual average concentration is $\mu = 8.1\%$, calculate the probability of committing a Type II error, denoted as $\beta(8.1)$. (10 points)

          An industrial chemist is checking for the desired concentration of a certain chemical component in a batch of polymer blend. The target concentration is set at 8.2%. To verify whether the actual average concentration matches the target for the current production line using a significance level of $\alpha = 0.05$, 15 independently obtained samples are taken for analysis. Assume that the concentration of the chemical component in a sample is normally distributed with $\sigma = 0.5\%$ and that the sample mean is $\bar{x} = 8.0$.

1. Based on the information provided, perform a hypothesis test to determine if there is statistical evidence that the true average concentration of the chemical component is different from 8.2%. State the null and alternative hypotheses and calculate the P-value. Based on the P-value and the significance level, explain your conclusion. (10 points)

2. Assuming that the actual average concentration is $\mu = 8.1\%$, calculate the probability of committing a Type II error, denoted as $\beta(8.1)$. (10 points)

$An industrial chemist is checking for the desired concentration of a certain chemical component in a batch of polymer blend. The target concentration is set at 8.2%. To verify whether the actual average concentration matches the target for the current production line using a significance level of $\alpha = 0.05$, 15 independently obtained samples are taken for analysis. Assume that the concentration of the chemical component in a sample is normally distributed with $\sigma = 0.5\%$ and that the sample mean is $\bar{x} = 8.0$. 1. Based on the information provided, perform a hypothesis test to determine if there is statistical evidence that the true average concentration of the chemical component is different from 8.2%. State the null and alternative hypotheses and calculate the P-value. Based on the P-value and the significance level, explain your conclusion. (10 points) 2. Assuming that the actual average concentration is μ = 8.1%, calculate the probability of committing a Type II error, denoted as β(8.1). (10 points)$

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