A sample of size 6 will be drawn from a normal population with mean 61 and standard deviation 14. Use the TI-84 Plus calculator. Is it appro

Question

A sample of size 6 will be drawn from a normal population with mean 61 and standard deviation 14. Use the TI-84 Plus calculator. Is it appropriate to use the normal distribution to find probabilities for x?

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Mary 1 week 2021-11-17T14:28:05+00:00 1 Answer 0 views 0

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    2021-11-17T14:29:58+00:00

    Answer:

    X \sim N(61,14)  

    Where \mu=61 and \sigma=14

    Since the distribution for X is normal then the distribution for the sample mean  is also normal and given by:

    \bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})

     \mu_{\bar X} = 61

    \sigma_{\bar X}= \frac{14}{\sqrt{6}}= 5.715

    So then is appropiate use the normal distribution to find the probabilities for \bar X

    Step-by-step explanation:

    Previous concepts

    Normal distribution, is a “probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean”.

    The Z-score is “a numerical measurement used in statistics of a value’s relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean”.  The letter \phi(b) is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words: \phi(b)=P(z<b)

    Solution to the problem

    Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

    X \sim N(61,14)  

    Where \mu=61 and \sigma=14

    Since the distribution for X is normal then the distribution for the sample mean \bar X is also normal and given by:

    \bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})

     \mu_{\bar X} = 61

    \sigma_{\bar X}= \frac{14}{\sqrt{6}}= 5.715

    So then is appropiate use the normal distribution to find the probabilities for \bar X

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