High School

Some sources report that the weights of full-term newborn babies in a certain town have a mean of 99 pounds and a standard deviation of 1.2 pounds and are Normally distributed.

a. What is the probability that one newborn baby will have a weight within 1.2 pounds of the mean—that is, between 97.8 and 100.2 pounds, or within one standard deviation of the mean?

b. What is the probability that the average of nine babies' weights will be within 1.2 pounds of the mean; will be between 97.8 and 100.2 pounds?

c. Explain the difference between (a) and (b).

Answer :

To address this question, let's first understand the concept of normal distribution and standard deviation. When data follows a normal distribution, we can determine probabilities using the properties of that distribution.

Part (a): Probability for One Newborn Baby

  1. Understanding the Question:

    • The mean weight of newborn babies is given as [tex]\mu = 99[/tex] pounds.
    • The standard deviation is [tex]\sigma = 1.2[/tex] pounds.
    • We need to find the probability that a baby's weight is between 97.8 and 100.2 pounds. This range represents one standard deviation from the mean.

  2. Using the Empirical Rule:

    • According to the empirical rule (68-95-99.7 rule) for normal distribution:
      • Approximately 68% of data within a normal distribution falls within one standard deviation from the mean.
    • Thus, the probability that one newborn baby's weight will be within 1.2 pounds of the mean is about 68%.

Part (b): Probability for the Average of Nine Babies

  1. Understanding the New Scenario:

    • We now consider the average weight of nine newborn babies.

  2. Central Limit Theorem (CLT) Application:

    • According to the Central Limit Theorem, the distribution of the sample mean will be normally distributed with:
      • Mean ([tex]\mu_{\text{sample}}[/tex]) equal to the population mean, [tex]\mu = 99[/tex] pounds.
      • Standard deviation ([tex]\sigma_{\text{sample}}[/tex]) equal to the population standard deviation divided by the square root of the sample size. Thus, [tex]\sigma_{\text{sample}} = \frac{1.2}{\sqrt{9}} = 0.4[/tex] pounds.

  3. Calculating the Probability:

    • We need to find the probability that the average weight of these nine babies is between 97.8 and 100.2 pounds.
    • Using the empirical rule for one standard deviation again for the sample mean:
      • The probability remains about 68% that the average weight falls within this range due to the sample size.

Part (c): Difference Between (a) and (b)

  1. Nature of the Data: Individual vs. Sample Average:

    • Part (a) deals with the weight of a single baby, while Part (b) deals with the average weight of nine babies.

  2. Impact of Sample Size:

    • In Part (b), the variability (or standard deviation) of the average is smaller due to the larger sample size (sample size = 9) by a factor of [tex]\sqrt{n}[/tex].

  3. Conceptual Understanding:

    • The reduction in variability due to averaging is why even with a smaller standard deviation (0.4 pounds) in Part (b), the probability of staying within 1.2 pounds (or 3 standard deviations in the context of the sample distribution) remains significant.

These concepts illustrate key principles of probability and statistics, such as normal distribution, the empirical rule, and the central limit theorem, all of which provide tools for understanding variations and patterns in data.

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