In this exercise you will test out the assumptions of the central limit theorem.

Question

In this exercise you will test out the assumptions of the central limit theorem. This theorem states that when you find the mean of all of the possible samples of a certain size from a population, the distribution of the sample means will be normally distributed, regardless of the original distribution of the data. The data for this exercise comes from a "penny-trap". Pennies received as change were placed into a box for a period of 6 months, then counted and the dates on the pennies noted. The workbook for this exercise can be downloaded here.

Using this data you will:

Calculate the mean, range and standard deviation of the dates on the pennies.

Make a frequency distribution of the dates on the pennies.

Use that frequency distribution to make a histogram of the data.

Generate 100 random samples each of size 10, 20, and 40

Calculate the mean of each sample and make a frequency distribution for the samples.

Use the frequency distribution to make a histogram of the data.

Calculate the mean, range and standard deviation of each of the sampling distributions.

Answer the following questions in the provided box:

How does the shape of your sampling distributions compare to the original data set and each other?

Do the mean, range and standard deviation of the sample means change as the sample size changes? If so, how?

Here is the Data being used, if you could please leave excel formulas! :

2014 1977 1977 1991 1997 2011 1990 2006 1964 2003 1964 2011 2015 2016 2011 2015 1996 1980 1974 1994 1975 2015 1999 2016 1980 2010 2016 1995 2005 2013 1996 2013 1981 1975 2015 2000 1989 1978 2016 2011 2015 1980 1966 2014 1996 2007 2004 2002 2015 1971 2010 1990 2014 2010 2008 1994 1975 2015 1975 1975 1985 2015 2015 1990 1974 1994 2013 2014 2012 2014 1997 2015 1985 2011 1992 1993 1980 2015 1984 1996 2015 1997 1988 2015 1992 1972 1983 1997 1984 1983 2014 2014 1995 1998 1978 1997 1977 2014 1982 2014 2014 2015 1993 2011 2013 2015 1980 2016 2003 2013 2014 2011 1988 1983 1983 1982 1999 2016 1965 1997 1976 2000 2014 1999 1984 2015 1988 1965 2006 2013 2015 2014 1983 2015 2009 2013 1964 2012 2003 1989 1989 2015 2015 1989 2013 2000 2001 2012 1982 2007 2008 1985 1995 1997 1991 1985 1997 1982 2009 1969 2000 2013 2008 2016 2000 1982 1974 1972 2013 2015 1974 2002 2000 2012 1985 2014 2006 1987 2015 2002 2015 2015 1999 2016 2007 2013 2014 1997 1981 1964 1993 2006 1992 1997 1967 1982 1973 2013 1982 2004 1966 2000 2008 2015 1961 1959 2013 1985 2015 2016 2015 1998 2016 1958 2015 1995 1991 1975 1981 1987 2002 1995 1964 1978 2016 1987 2015 1998 2010 2006 2016 1989 2014 2008 2013 2016 2013 2014 1975 2016 2001 1975 2006 1992 1992 1980 1971 1995 1999 2003 2006 2008 1985 2013 2012 2016 2005 2004 1979 1981 1982 2009 2004 2000 1996 2000 2012 2011 2013 2013 1975 1979 1979 1995 1998 1989 2011 1974 1979 2012 1983 2015 2009 1999 2014 2014 2016 1992 2014 2012 2001 1979 2013 2014 2005 1979 2000 1993 1992 1984 1968 2014 2007 1977 2015 2010 1999 2014 1964 2011 1992 2015 2016 1966 2014 2012 2015 2011 2008 2008 2016 1982 2008 1998 2011 2013 2015 1987 2015 1982 2015 1982 1980 1999 1996 2012 1994 2014 1985 2015 2006 2006 1981 1985 2015 1982 1964 1982 2015 1975 1961 2010 1997 1961

Explanation / Answer

We use R-software

rm(list=ls(all=TRUE))
x=c(2014.00,
1977.00,
1977.00,
1991.00,
1997.00,
2011.00,
1990.00,
2006.00,
1964.00,
2003.00,
1964.00,
2011.00,
2015.00,
2016.00,
2011.00,
2015.00,
1996.00,
1980.00,
1974.00,
1994.00,
1975.00,
2015.00,
1999.00,
2016.00,
1980.00,
2010.00,
2016.00,
1995.00,
2005.00,
2013.00,
1996.00,
2013.00,
1981.00,
1975.00,
2015.00,
2000.00,
1989.00,
1978.00,
2016.00,
2011.00,
2015.00,
1980.00,
1966.00,
2014.00,
1996.00,
2007.00,
2004.00,
2002.00,
2015.00,
1971.00,
2010.00,
1990.00,
2014.00,
2010.00,
2008.00,
1994.00,
1975.00,
2015.00,
1975.00,
1975.00,
1985.00,
2015.00,
2015.00,
1990.00,
1974.00,
1994.00,
2013.00,
2014.00,
2012.00,
2014.00,
1997.00,
2015.00,
1985.00,
2011.00,
1992.00,
1993.00,
1980.00,
2015.00,
1984.00,
1996.00,
2015.00,
1997.00,
1988.00,
2015.00,
1992.00,
1972.00,
1983.00,
1997.00,
1984.00,
1983.00,
2014.00,
2014.00,
1995.00,
1998.00,
1978.00,
1997.00,
1977.00,
2014.00,
1982.00,
2014.00,
2014.00,
2015.00,
1993.00,
2011.00,
2013.00,
2015.00,
1980.00,
2016.00,
2003.00,
2013.00,
2014.00,
2011.00,
1988.00,
1983.00,
1983.00,
1982.00,
1999.00,
2016.00,
1965.00,
1997.00,
1976.00,
2000.00,
2014.00,
1999.00,
1984.00,
2015.00,
1988.00,
1965.00,
2006.00,
2013.00,
2015.00,
2014.00,
1983.00,
2015.00,
2009.00,
2013.00,
1964.00,
2012.00,
2003.00,
1989.00,
1989.00,
2015.00,
2015.00,
1989.00,
2013.00,
2000.00,
2001.00,
2012.00,
1982.00,
2007.00,
2008.00,
1985.00,
1995.00,
1997.00,
1991.00,
1985.00,
1997.00,
1982.00,
2009.00,
1969.00,
2000.00,
2013.00,
2008.00,
2016.00,
2000.00,
1982.00,
1974.00,
1972.00,
2013.00,
2015.00,
1974.00,
2002.00,
2000.00,
2012.00,
1985.00,
2014.00,
2006.00,
1987.00,
2015.00,
2002.00,
2015.00,
2015.00,
1999.00,
2016.00,
2007.00,
2013.00,
2014.00,
1997.00,
1981.00,
1964.00,
1993.00,
2006.00,
1992.00,
1997.00,
1967.00,
1982.00,
1973.00,
2013.00,
1982.00,
2004.00,
1966.00,
2000.00,
2008.00,
2015.00,
1961.00,
1959.00,
2013.00,
1985.00,
2015.00,
2016.00,
2015.00,
1998.00,
2016.00,
1958.00,
2015.00,
1995.00,
1991.00,
1975.00,
1981.00,
1987.00,
2002.00,
1995.00,
1964.00,
1978.00,
2016.00,
1987.00,
2015.00,
1998.00,
2010.00,
2006.00,
2016.00,
1989.00,
2014.00,
2008.00,
2013.00,
2016.00,
2013.00,
2014.00,
1975.00,
2016.00,
2001.00,
1975.00,
2006.00,
1992.00,
1992.00,
1980.00,
1971.00,
1995.00,
1999.00,
2003.00,
2006.00,
2008.00,
1985.00,
2013.00,
2012.00,
2016.00,
2005.00,
2004.00,
1979.00,
1981.00,
1982.00,
2009.00,
2004.00,
2000.00,
1996.00,
2000.00,
2012.00,
2011.00,
2013.00,
2013.00,
1975.00,
1979.00,
1979.00,
1995.00,
1998.00,
1989.00,
2011.00,
1974.00,
1979.00,
2012.00,
1983.00,
2015.00,
2009.00,
1999.00,
2014.00,
2014.00,
2016.00,
1992.00,
2014.00,
2012.00,
2001.00,
1979.00,
2013.00,
2014.00,
2005.00,
1979.00,
2000.00,
1993.00,
1992.00,
1984.00,
1968.00,
2014.00,
2007.00,
1977.00,
2015.00,
2010.00,
1999.00,
2014.00,
1964.00,
2011.00,
1992.00,
2015.00,
2016.00,
1966.00,
2014.00,
2012.00,
2015.00,
2011.00,
2008.00,
2008.00,
2016.00,
1982.00,
2008.00,
1998.00,
2011.00,
2013.00,
2015.00,
1987.00,
2015.00,
1982.00,
2015.00,
1982.00,
1980.00,
1999.00,
1996.00,
2012.00,
1994.00,
2014.00,
1985.00,
2015.00,
2006.00,
2006.00,
1981.00,
1985.00,
2015.00,
1982.00,
1964.00,
1982.00,
2015.00,
1975.00,
1961.00,
2010.00,
1997.00,
1961.00
)

mean(x)
1997.975
sd(x)
15.87975
range(x)
1958 2016
breaks=seq(1955, 2020, by=5)
y=cut(x, breaks, right=FALSE)
freq =cbind(table(y))

          [,1]
[1955,1960)    2
[1960,1965)   10
[1965,1970)    8
[1970,1975)   10
[1975,1980)   24
[1980,1985)   35
[1985,1990)   22
[1990,1995)   22
[1995,2000)   37
[2000,2005)   25
[2005,2010)   29
[2010,2015)   73
[2015,2020)   57

hist(c(freq), right=FALSE)

y=list()
m=c();s=c();r=c()
for( i in 1:100){
y[[i]]=sample(x, 10)
m[i]=mean(y[[i]])
s[i]=sd(y[[i]])
r[[i]]=range(y[[i]])
}

> m
[1] 2006.7 1998.2 2000.3 1992.1 2003.4 1999.1 1997.0 2002.6 1998.6 2007.1
[11] 1999.1 1999.1 2005.9 2002.5 2007.6 1996.3 2002.1 1998.2 2000.9 1999.4
[21] 2001.4 2001.6 1992.0 2001.3 2001.5 1997.3 1998.8 1995.2 1996.1 1995.2
[31] 1997.3 1997.1 1994.6 2001.3 1994.5 1996.2 1995.8 1987.9 1992.2 1994.0
[41] 1999.5 1995.3 1988.2 1996.9 1995.4 2005.7 2001.0 2001.8 2003.5 2000.2
[51] 1999.5 1999.5 2001.1 1996.7 1994.2 2000.2 1998.7 1993.6 1992.0 1992.3
[61] 2008.3 1993.8 1999.9 1997.5 2001.3 1995.9 1993.8 1984.0 2000.8 1991.2
[71] 1998.2 1990.0 1984.4 1990.7 1993.5 1995.7 2000.6 2001.4 1996.9 1999.1
[81] 2000.2 1999.0 1999.0 1991.1 1996.8 2000.4 1991.2 1998.2 2001.8 2001.7
[91] 1984.8 1988.7 1994.0 1996.7 1992.2 1999.0 2000.7 1999.6 1994.7 2003.6
> s
[1] 10.541453 16.857573 13.416822 19.416202 15.283979 19.829551 14.636332
[8] 12.203825 11.739771 9.620002 14.278578 17.329807 11.090036 16.297580
[15] 6.719788 18.288126 15.523817 16.988885 15.095621 13.384900 11.964763
[22] 16.036070 12.283684 15.122096 13.083068 13.416822 14.543422 16.369348
[29] 13.995634 19.042642 18.037307 17.916163 18.596296 14.560601 19.934615
[36] 15.746605 18.280530 14.510150 17.831308 16.295875 14.953632 18.024983
[43] 19.492449 16.360182 18.626146 10.414200 16.213849 14.838389 15.897239
[50] 17.862437 14.245857 15.615164 12.922418 19.647165 17.254951 11.321563
[57] 19.618868 18.939670 14.514361 17.701538 9.764448 20.573986 16.475234
[64] 17.251409 12.640763 15.940514 20.181400 17.346149 10.706177 20.307634
[71] 18.954917 21.044925 18.512458 15.951663 20.828666 18.061930 16.331292
[78] 14.198591 18.621672 18.333030 17.216271 17.211108 12.806248 18.417685
[85] 15.788885 14.546477 14.672726 15.901083 19.176374 17.378467 15.490499
[92] 13.416822 13.498971 15.173442 20.842265 11.832160 13.936762 18.974837
[99] 13.140692 12.773236
> r
[[1]]
[1] 1982 2016

[[2]]
[1] 1961 2014

[[3]]
[1] 1977 2015

[[4]]
[1] 1964 2014

[[5]]
[1] 1968 2016

[[6]]
[1] 1961 2016

[[7]]
[1] 1977 2015

[[8]]
[1] 1984 2016

[[9]]
[1] 1982 2016

[[10]]
[1] 1984 2015

[[11]]
[1] 1975 2016

[[12]]
[1] 1964 2016

[[13]]
[1] 1980 2016

[[14]]
[1] 1967 2016

[[15]]
[1] 1996 2015

[[16]]
[1] 1966 2016

[[17]]
[1] 1980 2016

[[18]]
[1] 1961 2016

[[19]]
[1] 1976 2015

[[20]]
[1] 1981 2015

[[21]]
[1] 1980 2015

[[22]]
[1] 1966 2016

[[23]]
[1] 1978 2015

[[24]]
[1] 1973 2015

[[25]]
[1] 1980 2015

[[26]]
[1] 1975 2014

[[27]]
[1] 1979 2016

[[28]]
[1] 1964 2015

[[29]]
[1] 1974 2015

[[30]]
[1] 1966 2015

[[31]]
[1] 1974 2015

[[32]]
[1] 1969 2016

[[33]]
[1] 1964 2014

[[34]]
[1] 1974 2015

[[35]]
[1] 1961 2015

[[36]]
[1] 1975 2016

[[37]]
[1] 1964 2016

[[38]]
[1] 1964 2011

[[39]]
[1] 1961 2015

[[40]]
[1] 1966 2015

[[41]]
[1] 1974 2015

[[42]]
[1] 1965 2015

[[43]]
[1] 1958 2015

[[44]]
[1] 1975 2016

[[45]]
[1] 1964 2013

[[46]]
[1] 1989 2016

[[47]]
[1] 1973 2016

[[48]]
[1] 1980 2015

[[49]]
[1] 1978 2016

[[50]]
[1] 1971 2015

[[51]]
[1] 1981 2016

[[52]]
[1] 1975 2016

[[53]]
[1] 1981 2015

[[54]]
[1] 1967 2015

[[55]]
[1] 1974 2016

[[56]]
[1] 1979 2015

[[57]]
[1] 1967 2016

[[58]]
[1] 1965 2015

[[59]]
[1] 1971 2014

[[60]]
[1] 1971 2015

[[61]]
[1] 1983 2015

[[62]]
[1] 1958 2016

[[63]]
[1] 1964 2015

[[64]]
[1] 1964 2014

[[65]]
[1] 1980 2016

[[66]]
[1] 1975 2016

[[67]]
[1] 1961 2015

[[68]]
[1] 1961 2013

[[69]]
[1] 1989 2016

[[70]]
[1] 1964 2015

[[71]]
[1] 1959 2016

[[72]]
[1] 1958 2015

[[73]]
[1] 1961 2015

[[74]]
[1] 1964 2011

[[75]]
[1] 1964 2016

[[76]]
[1] 1971 2016

[[77]]
[1] 1977 2015

[[78]]
[1] 1977 2015

[[79]]
[1] 1959 2014

[[80]]
[1] 1964 2015

[[81]]
[1] 1974 2016

[[82]]
[1] 1966 2016

[[83]]
[1] 1979 2015

[[84]]
[1] 1961 2015

[[85]]
[1] 1961 2014

[[86]]
[1] 1975 2016

[[87]]
[1] 1974 2013

[[88]]
[1] 1972 2015

[[89]]
[1] 1964 2016

[[90]]
[1] 1966 2016

[[91]]
[1] 1961 2015

[[92]]
[1] 1965 2015

[[93]]
[1] 1975 2015

[[94]]
[1] 1966 2016

[[95]]
[1] 1961 2016

[[96]]
[1] 1977 2014

[[97]]
[1] 1975 2015

[[98]]
[1] 1958 2016

[[99]]
[1] 1972 2014

[[100]]
[1] 1982 2015

Change the sample from 20 and 40

the mean, range and standard deviation of the sample means change as the sample size changes because it goes to true parameter.

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