\"Productivity\" is a term you hear quite often in news reports about the econom

Question

"Productivity" is a term you hear quite often in news reports about the economy. Productivity isnt everything, but in the long run it is almost everything. A countrys ability to improve its standard of living over time depends almost entirely on its ability to raise its output per worker.

Productivity measures how efficiently production inputs, such as labor and capital, are being used in an economy to produce a given level of output. Knowledgeable, skilled workers with the proper tools such as computers or manufacturing equipment result in high productivity.

This Productivity Excel file shows the percent change in worker output per hour from the previous quarter for the United States from 1988 through 2008.

Question 2. These data can be approximated quite well by a N(3.4, 3.1) model. Economists become alarmed when productivity decreases. According to the normal model what is the probability that the percent change in worker output per hour from the previous quarter is more than 1.3 standard deviations below the mean?

Question 3. What is the probability that the percent change in worker output from the previous quarter is between -1.25 and 3.555? Use the normal model mentioned at the beginning of question 2.

Data:

Year Period Value 1988 Qtr1 -0.6 1988 Qtr2 1.8 1988 Qtr3 0.7 1988 Qtr4 3.6 1989 Qtr1 2.7 1989 Qtr2 -2.5 1989 Qtr3 -1.8 1989 Qtr4 3 1990 Qtr1 5.1 1990 Qtr2 1 1990 Qtr3 4.9 1990 Qtr4 -0.3 1991 Qtr1 0.5 1991 Qtr2 5.5 1991 Qtr3 6.5 1991 Qtr4 0.4 1992 Qtr1 2.2 1992 Qtr2 6.1 1992 Qtr3 7.8 1992 Qtr4 -0.4 1993 Qtr1 3.4 1993 Qtr2 0.2 1993 Qtr3 0.7 1993 Qtr4 5.6 1994 Qtr1 3.2 1994 Qtr2 5.3 1994 Qtr3 2.1 1994 Qtr4 5.3 1995 Qtr1 6.2 1995 Qtr2 4.2 1995 Qtr3 3.4 1995 Qtr4 3.5 1996 Qtr1 5.4 1996 Qtr2 0.6 1996 Qtr3 5 1996 Qtr4 3 1997 Qtr1 5.7 1997 Qtr2 6.8 1997 Qtr3 9 1997 Qtr4 7 1998 Qtr1 4.5 1998 Qtr2 2.7 1998 Qtr3 6.3 1998 Qtr4 3.3 1999 Qtr1 6.4 1999 Qtr2 3.6 1999 Qtr3 -0.4 1999 Qtr4 11.1 2000 Qtr1 4 2000 Qtr2 2.9 2000 Qtr3 0.3 2000 Qtr4 2.4 2001 Qtr1 -0.6 2001 Qtr2 3.1 2001 Qtr3 1.3 2001 Qtr4 7.9 2002 Qtr1 9.4 2002 Qtr2 8.7 2002 Qtr3 7 2002 Qtr4 3.7 2003 Qtr1 8.8 2003 Qtr2 4.9 2003 Qtr3 8.9 2003 Qtr4 -1.5 2004 Qtr1 -1.2 2004 Qtr2 3.4 2004 Qtr3 3.3 2004 Qtr4 7.3 2005 Qtr1 6.7 2005 Qtr2 4.9 2005 Qtr3 2.1 2005 Qtr4 0.2 2006 Qtr1 -1.4 2006 Qtr2 0.5 2006 Qtr3 4.7 2006 Qtr4 1.7 2007 Qtr1 3.1 2007 Qtr2 4.5 2007 Qtr3 6 2007 Qtr4 2.6 2008 Qtr1 2.9 2008 Qtr2 -1.1 2008 Qtr3 -2.2 2008 Qtr4 -4

Explanation / Answer

Here the given normal model is X ~ Normal (3.4, 3.1)

here as per normal model, we have to find that the probability that the percent change in worker output per hour from the previous quarter is more than 1.3 standard deviations below the mean.

Pr(X < - 1.3 )

Z = 1.3 here

Pr(X < - 1.3 ) = 0.0968

Question2

Here , Pr(-1.25 < X < 3.555) = NORMAL (X < 3.555 ; 3.4; 3.1) - NORMAL (X < -1.25 ; 3.4 ; 3.1) = (Z2) - (Z1)

where is the standard normal cumulative distribution

Z2 = (3.555 - 3.4)/3.1 = 0.05, Z1 = (-1.25 - 3.4)/ 3.1 = -1.5

Pr(-1.25 < X < 3.555) = NORMAL (X < 3.555 ; 3.4; 3.1) - NORMAL (X < -1.25 ; 3.4 ; 3.1) = (Z2) - (Z1)

= (0.05) - (-1.5)

= 0.5199 - 0.0668

= 0.4531

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