What is the probability that our sample will have a mean between 195 lbs and 200 lbs?

The probability of a sample mean falling within a specific range can be calculated using the standard normal distribution and z-scores, provided the population mean, standard deviation, and sample size are known. This typically involves converting the range of sample mean values to z-scores and finding the corresponding area under the standard normal curve.

Explanation:

The probability that a sample will have a mean between 195 lbs and 200 lbs can be determined using the concepts of probability and the properties of a normal distribution if the underlying population distribution is normal. Given a population with a known mean and standard deviation, and the sample size, one can use the standard normal distribution (z-scores) to find this probability. According to the Central Limit Theorem, the sampling distribution of the sample mean is normally distributed, even if the population distribution is not normal, given that the sample size is sufficiently large (usually n > 30).

As an example, we can consider a group of samples where the population mean is 90 and the standard deviation is 15, with the sample size n = 25. To find the probability that the sample mean is between 85 and 92, we would convert these values into z-scores and use the standard normal distribution to calculate the probabilities associated with these z-scores. This process involves subtracting the population mean from the sample mean, dividing by the standard error (standard deviation divided by the square root of the sample size), and then using a z-table or statistical software to find the area under the standard normal curve between these z-scores.

Furthermore, certain confidence intervals, like 95% or 90%, can typically be associated with a certain number of standard deviations from the mean in the normal distribution. The exact probability in each case will depend on the details provided, such as the population mean, the standard deviation, the size of the sample, and the range of sample mean values in question.

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