Let Y 1 , Y 2 , . . . , Y n be random samples from a normal distribution where the mean is 2 and the variance is 4. How large must n be in order that P ( 1.9 ≤ Y ¯ ≤ 2.1 ) ≥ 0.99?My attemp^We are computing for the sample mean of our random sample that was given in the problem. By definition, z = Y ¯ − μ σ n . So rewrite the equation so we can transform the data to make the mean 0 and the standard deviation 1. If I do that, I get, P ( 1.9 − 2 2 n ≤ z ≤ 2.1 − 2 2 n ) ≥ 0.99. This means that P ( − 0.1 2 n ≤ z ≤ 0.1 2 n ) ≥ 0.99. This also means that P ( − 0.05 n ≤ z ≤ 0.05 n ) ≥ 0.99. I am not really sure what to do after this step. I am trying to use the definition of the normal distribution, however that was too difficult to do.Do you guys know what to do after this step?

Question

Let       Y          1        ,      Y          2        ,  .  .  .  ,      Y          n       be random samples from a normal distribution where the mean is 2 and the variance is 4. How large must n be in order that   P  (  1.9  ≤      Y    ¯    ≤  2.1  )  ≥  0.99?My attemp^We are computing for the sample mean of our random sample that was given in the problem. By definition,   z  =                    Y        ¯            −      μ              σ              n            . So rewrite the equation so we can transform the data to make the mean 0 and the standard deviation 1. If I do that, I get,  P  (            1.9      −      2              2              n              ≤  z  ≤            2.1      −      2              2              n              )  ≥  0.99. This means that  P  (            −      0.1              2              n              ≤  z  ≤      0.1          2              n              )  ≥  0.99. This also means that  P  (  −  0.05      n    ≤  z  ≤  0.05      n    )  ≥  0.99. I am not really sure what to do after this step. I am trying to use the definition of the normal distribution, however that was too difficult to do.Do you guys know what to do after this step?

Braxton Gallagher · Accepted Answer

You just have to set  0.05      n    ≥      Φ          −      1        (  0.995  )and solve for   n

Let Y <mrow class="MJX-TeXAtom-ORD"> 1 </mrow> </msub> , Y

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