High School

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Assuming that the true population mean is 98.6 degrees F and the population standard deviation is 0.62 degrees F, what is the probability of obtaining a sample of 106 temperatures with a sample mean of 98.2 degrees F or less?

Answer :

The probability of obtaining a sample mean of 98.2 degrees Fahrenheit or less from a population with a mean of 98.6 and standard deviation of 0.62, in a sample of 106 temperatures, is practically zero due to the Z-score being extremely low.

The student asked about the probability of obtaining a sample mean of 98.2 degrees fahrenheit or less, given the true population mean is 98.6 degrees with a standard deviation of 0.62, using a sample of 106 temperatures. To find this probability, we can use the standard normal distribution or Z-score formula, given by:

Z = (X - μ) / (σ / √n)

Where:

  • X is the sample mean
  • μ is the population mean
  • σ is the population standard deviation
  • n is the sample size

We calculate the Z-score as follows:

Z = (98.2 - 98.6) / (0.62 / √106) = -4.6 / (0.62 / 10.2956) = -4.6 / 0.0601 ≈ -76.54

This Z-score is extremely low, indicating that the sample mean of 98.2 is very far below the population mean, thus the probability of getting this sample mean or lower is practically zero.

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