Question Suppose heights , in inches, of orangutans are normally distributed and have a known population standard deviation of 4 inches and an unknown population mean . A random sample of 16 orangutans is taken and gives a sample mean of 56 inches. Find the confidence interval for the population mean with a 95% confidence level. You may use a calculator or the common z values above. Round all numbers ; to three decimal places if necessary. Provide your answer below: (square ,square )

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Question Suppose heights , in inches, of orangutans are normally distributed and have a known population standard deviation of 4 inches and an unknown population mean . A random sample of 16 orangutans is taken and gives a sample mean of 56 inches. Find the confidence interval for the population mean with a 95% confidence level. You may use a calculator or the common z values above. Round all numbers ; to three decimal places if necessary. Provide your answer below: (square ,square )

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To find the confidence interval for the population mean with a 95% confidence level, we will use the sample mean, the population standard deviation, and the appropriate z-value for a 95% confidence level.Step 1: Identify the given information.The sample size (n) is 16.The population standard deviation (σ) is 4 inches.The sample mean (\(\bar{x}\)) is 56 inches.For a 95% confidence interval, the z-value (\(z_{0.025}\)) is 1.960 (since 95% confidence level corresponds to a 2.5% significance level in each tail of the normal distribution).Step 2: Calculate the standard error of the mean (SEM).The standard error of the mean is calculated by dividing the population standard deviation by the square root of the sample size.\[SEM = \frac{\sigma}{\sqrt{n}}\]\[SEM = \frac{4}{\sqrt{16}}\]\[SEM = \frac{4}{4}\]\[SEM = 1\]Step 3: Calculate the margin of error (ME).The margin of error is calculated by multiplying the z-value by the standard error of the mean.\[ME = z \cdot SEM\]\[ME = 1.960 \cdot 1\]\[ME = 1.960\]Step 4: Calculate the confidence interval.The confidence interval is calculated by adding and subtracting the margin of error from the sample mean.Lower limit (\(L\)) = \(\bar{x} - ME\)Upper limit (\(U\)) = \(\bar{x} + ME\)\[L = 56 - 1.960\]\[L = 54.040\]\[U = 56 + 1.960\]\[U = 57.960\]Step 5: Round the results to three decimal places.\[L \approx 54.040\]\[U \approx 57.960\]Therefore, the 95% confidence interval for the population mean height of orangutans is approximately (54.040, 57.960) inches.**Accurate Answer**: The 95% confidence interval for the population mean is (54.040, 57.960) inches.

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