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The birth weights of newborn babies in the United States follow a normal distribution with a mean of 3.4 kg and a standard deviation of 0.6 kg. Researchers interested in studying how children gain weight decide to take random samples of 100 newborn babies and calculate the sample mean birth weight for each sample. Calculate the mean and standard deviation of the sampling distribution

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Answer:

3.4 kg

0.06 kg

Step-by-step explanation:

Given that:

Birthweight follows a normal distribution :

Mean weighf (m) = 3.4 kg

Standard deviation (σ) = 0.6 kg

Sample size (n) = 100 new born babies

calculate the sample mean birth weight for each sample. Calculate the mean and standard deviation of the sampling distribution

The sample mean birth weight = population mean birth weight for a normal distribution according to the central Limit theorem.

Hence, sample mean birth weight = 3.4kg

Standard deviation of sampling distribution = standard error = σ/√n

0.6 / √100

= 0.6 / 10

= 0.06 kg

Using the Central Limit Theorem, it is found that:

  • The mean is of 3.4 kg.
  • The standard deviation is of 0.06 kg.

Central Limit Theorem

  • The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
  • For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem:

  • Mean of 3.4 kg, hence [tex]\mu = 3.4[/tex].
  • Standard deviation of 0.6 kg, hence [tex]\sigma = 0.6[/tex].
  • Samples of 100, hence [tex]n = 100, s = \frac{0.6}{\sqrt{100}} = 0.06[/tex]

Hence, the mean of the sampling distribution is of 3.4 kg and the standard deviation is of 0.06 kg.

To learn more about the Central Limit Theorem, you can take a look at https://brainly.com/question/24663213

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