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A student was asked to find a 95% confidence interval for weight of their backpacks in pounds using data from a random sample of size n = 20. Which of the following is a correct interpretation of the interval 3.1 < μ < 8.6? Assume the population is normally distributed.With 95% confidence, the weight of a randomly selected backpack will be between 3.1 and 8.6 pounds.There is a 95% chance that the mean of a sample of 20 backpacks will weigh between 3.1 and 8.6 pounds.There is a 95% chance that the weight is between 3.1 and 8.6.With 95% confidence, the mean weight of all backpacks is between 3.1 and 8.6 pounds.The sample mean weight of all backpacks is between 3.1 and 8.6 pounds, 95% of the time. We know this is true because the mean of our sample is between 3.1 and 8.6.

Question

A student was asked to find a 95% confidence interval for weight of their backpacks in pounds using data from a random sample of size n = 20. Which of the following is a correct interpretation of the interval 3.1 < μ < 8.6? Assume the population is normally distributed.With 95% confidence, the weight of a randomly selected backpack will be between 3.1 and 8.6 pounds.There is a 95% chance that the mean of a sample of 20 backpacks will weigh between 3.1 and 8.6 pounds.There is a 95% chance that the weight is between 3.1 and 8.6.With 95% confidence, the mean weight of all backpacks is between 3.1 and 8.6 pounds.The sample mean weight of all backpacks is between 3.1 and 8.6 pounds, 95% of the time. We know this is true because the mean of our sample is between 3.1 and 8.6.

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Solution

To interpret the 95% confidence interval 3.1 < μ < 8.6 correctly, let's analyze each option step by step:

  1. With 95% confidence, the weight of a randomly selected backpack will be between 3.1 and 8.6 pounds.

    • This statement is incorrect. A confidence interval for the mean does not provide information about individual data points, but rather about the population mean.
  2. There is a 95% chance that the mean of a sample of 20 backpacks will weigh between 3.1 and 8.6 pounds.

    • This statement is incorrect. The confidence interval is about the population mean, not about the mean of future samples.
  3. There is a 95% chance that the weight is between 3.1 and 8.6.

    • This statement is incorrect. It is too vague and does not specify that it refers to the population mean.
  4. With 95% confidence, the mean weight of all backpacks is between 3.1 and 8.6 pounds.

    • This statement is correct. It accurately reflects the interpretation of a confidence interval for the population mean.
  5. The sample mean weight of all backpacks is between 3.1 and 8.6 pounds, 95% of the time. We know this is true because the mean of our sample is between 3.1 and 8.6.

    • This statement is incorrect. The confidence interval pertains to the population mean, not the sample mean, and it does not mean that 95% of the time the sample mean will fall within this range.

Therefore, the correct interpretation is: With 95% confidence, the mean weight of all backpacks is between 3.1 and 8.6 pounds.

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