Let X be the temperature in °C at which a certain chemical reaction takes place, and let Y be the temperature in °F (so Y = 1.8X + 32). (a) If the median of the X distribution is p, show that 1.8 + 32...

Let X be the temperature in °C at which a certain chemical reaction takes place, and let Y be the temperature in °F (so Y = 1.8X + 32).<br>(a) If the median of the X distribution is p, show that 1.8 + 32 is the median of the Y distribution.<br>P(Y s 1.8ũ + 32)<br>= P<br>Y<br>< 1.8 + 32<br>= P<br>Since i is the median of X, P(X sP) =<br>0.5<br>0.5. Thus, 1.8ũ + 32 is the median of Y.<br>(b) How is the 90th percentile of the Y distribution related to the 90th percentile of the X distribution?<br>ny(.9) = ( 1.8<br>n,(.9) + ( 32<br>Verify your conjecture.<br>P(Y s 1.87(.9) + 32)<br><1.87 .9) + 32<br>= Pl<br>X<br><ny(.9)<br>= P<br>Since n1.9) is the 90th percentile of X, P(X s n1.9)) = 0.9<br>. Thus, 1.8n 1.9) + 32 is the 90th percentile of Y.<br>(c) More generally, if Y = ax + b, how is any particular percentile of the Y distribution related to the corresponding percentile of the X distribution?<br>nAe) = (| 1.8<br>ndo) - (32<br>This is a result of the following.<br>P(Y s an Ap) + b) = P( Y<br>s anx(p) +<br>= P X<br>Since n (p) is the (100p)th percentile of X, P(X sn 0)) = p. Thus, an (p) + b is the (100p)th percentile of Y.<br>

Extracted text: Let X be the temperature in °C at which a certain chemical reaction takes place, and let Y be the temperature in °F (so Y = 1.8X + 32). (a) If the median of the X distribution is p, show that 1.8 + 32 is the median of the Y distribution. P(Y s 1.8ũ + 32) = P Y < 1.8="" +="" 32="P" since="" i="" is="" the="" median="" of="" x,="" p(x="" sp)="0.5" 0.5.="" thus,="" 1.8ũ="" +="" 32="" is="" the="" median="" of="" y.="" (b)="" how="" is="" the="" 90th="" percentile="" of="" the="" y="" distribution="" related="" to="" the="" 90th="" percentile="" of="" the="" x="" distribution?="" ny(.9)="(" 1.8="" n,(.9)="" +="" (="" 32="" verify="" your="" conjecture.="" p(y="" s="" 1.87(.9)="" +="" 32)=""><1.87 .9)="" +="" 32="Pl" x="">

Jun 01, 2022

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Let X be the temperature in °C at which a certain chemical reaction takes place, and let Y be the temperature in °F (so Y = 1.8X + 32). (a) If the median of the X distribution is p, show that 1.8 + 32...

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