Question

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A simple random sample of size n= 44 is obtained from a population with u = 61 and o = 19 (a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of x. (b) Assuming the normal model can be used, determine P(<65.3). (c) Assuming the normal model can be used, determine P(x262.4). (a) What must be true regarding the distribution of the population? O A. The population must be normally distributed. OB. The population must be normally distributed and the sample size must be large. OC. The sampling distribution must be assumed to be normal. OD. Since the sample size is large enough, the population distribution does not need to be normal. Assuming the normal model can be used, describe the sampling distribution 3. Choose the correct answer below. 19 O A. Approximately normal, with Hz = 61 and 3 = 744 44 O B. Approximately normal, with Hz = 61 and on 19 OC. Approximately normal, with H3 = 61 and 0 = 19 (b) P(X<65.3)=(Round to four decimal places as needed.) (c) P(X262.4) = (Round to four decimal places as needed.)

Answer 1

Solution :

Given that ,

a) correct option is = A

A) Approximately normal with

$\mu$_{$\bar x$} = 61

$\sigma$_{$\bar x$} = $\sigma$ / $\sqrt$ n = 19 / $\sqrt$ 44

b) P($\bar x$ < 65.3) = P(($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$ _{$\bar x$} < (65.3 - 61) / 19 / $\sqrt$ 44 )

= P(z < 1.50)

Using z table

= 0.9332

c) P($\bar x$$\geq$ 62.4) = 1 - P($\bar x$$\leq$ 62.4)

= 1 - P[($\bar x$ - $\mu$ _{$\bar x$} ) / $\sigma$_{$\bar x$}$\leq$ (62.4 - 61) / 19 / $\sqrt$ 44 ]

= 1 - P(z $\leq$ 0.49)   

= 1 - 0.6879

= 0.3121