Question

1) the distribution and histogram of individual penny dates for the entire class (this will be our population), Math/BSAD 2170 Sampling Distributions and Central Limit Theorem 2) the distribution and histogram of the means from samples of 5 pennies (this is called a sampling distribution with n 5), 3) the distribution and histogram of the means from samples of 10 pennies (a sampling distribution with n 10), and 4) the distribution and histogram of the means of each sample of 25 pennies from Data Activity 1. I will also post the mean and standard deviation for our population, μ and σ , and the mean and standard deviation for each sampling distributions, 14 and ơ. (Data Activity 2 Summary). You will need to answer some questions about these graphs, means, and standard deviations in the analysis post Reply to this analysis post, answering the following questions: 1) How do the shapes of the sampling distributions, for n distribution of the population? 5 and for n 10, compare to the 2) How do the means of the sampling distributions, ,compare to the mean of the population, μ? That is compare μ, for n 5 with 1, and compare μ,for n 10 with μ 3) How do the standard deviations of the sampling distributions, ơr , compare to the standard deviation of the population, o? That is, compare ơī for n-5 with σ, and compare ơr for n- 10 with σ 4) a) For your sample means with n S, how close are your five sample means to the population mean, A, using the z-scores for your sample means,i? Are any of your sample means "unusual"? (Explicitly state your answers, such as "None of my sample means are unusual." Or, "The dates 1921 and 1923 are unusual." A simple "Yes" or "No." answer is unacceptable. Always answer questions with a complete sentence!) b) For your sample means with n - 10, how close are your five sample means from the population mean, u, in terms of the z-scores for your sample means, Are any of your sample means "unusual"? c) How close is your original sample mean, x , i.e., the mean you calculated in Data Activity 1, with n-25, to the population mean, μ, in terms of the z-score for your sample mean, Is your sample mean "unusual"? Your sample mean for n = 25, . Its z-score, z x is your sample mean "unusuar? Your sample means, n 10 Your sample z- scores z- scores means, n = 5 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 5) Using your results and those of a few (at least three) other student results, compare the results of this Data Activity with the results of the Central Limit Theorem. Data Activity 2, Sample Means Graph 1. Graph of the dates submitted by 20 students and the instructor. (One student submitted 26 dates. We will use this as our population, N 526.) Dates of Pennies, N 526 dates frequency 1960-1964 18 140 120 1965-1969 15 100 1970-1974 21 80 1975-1979 34 980-1984 29 40 985-1989 32 990-1994 37 995 1999 44 1966.1 1982.7 Dates 19993 2015.9 2000-2004 4 2005-2009 43 ●1960-1964 . 1985-1969 . 1970-1974 . 1975-1979 . 1980-1984 . 1985-1989 2010- 2014 84 ●1990-1994·1995-1999 ● 2000-2004 ● 2005-2009 . 2010-2014 . 2015-2019 2015-2019 126 we will consider these 526 dates as our population, N-526, μ-1999.3 and σ. 16.6. Graph 2. For samples of size n-5, the sampling distribution of sample means. Sample Means, n 5 freguensy5 dates 960-1969 30 970-1974 1 25 1975-19791 20 1980-1984 15 1985-19892 10 1990-1994 16 995-1999 12 dates 2000-2004 29 005-2009 15 1960-1969 1970-19741975-1979 1980-198 2010 20146 ●1985-1989-1990-1994 ●1995-1 .2 .2004 2010-2014 2015-2019 2005-2009 2015-2019 -5-19994, the standard deviation for these 85 samples The mean of these 85 sample means is = 7.9 means is 25, the sampling distribution of means, Graph 4 . For the 24 samples of size n 25 Sample Means, n dates frequency 12 1960-1969 30 970-1974 1975-1979 1980 1984 1985-1989 1990 1994 995-1999 1 2000- 2004 Dates 2005-2009 1960-1969 1970-3974 m 1975-19791980-1984 n 1985-1989 1990-199 2000-2004 12005-2009 2010-2014 2015-2019 2010 2014 1995-1999 2015-2019 The mean of these 24 sample means is a1999.5,the standard deviation for these 24 samples means is S2-5.3 Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 2 4 S.No. Penny Value 2000 1974 1974 1982 2005 1982 1982 1986 1986 2008 1982 2000 2018 2018 2018 2014 10 2014 2014 2014 1982 12 1982 1978 1978 13 14 2000 2014 2014 15 2014 1961 1961 16 2006 2006 2006 2006 17 2017 18 1986 1986 1986 19 1985 1985 1985 20 2000 2000 2000 21 2006 1982 2006 23 2017 2017 24 2014 2014 25 Mean1997.241994 2004.6 2001 1981.8 2005.4 S.No. Penny Value Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 2000 2000 2000 2000 2 19741974 1974 1974 1974 1982 2005 2005 1982 1982 1982 1982 1982 1986 1986 1986 19861986 2008 2008 2008 2008 2008 2008 2000 2000 2000 2000 2018 2018 2018 2014 2014 10 2014 2014 2014 2014 1982 1982 1982 12 1978 2000 2000 2000 2014 2014 2014 16 196 1961 17 2017 1986 19 1986 1985 1985 2000 23 982 2017 1982 2014 2014 Mean 1997.24 2000.1 1998.8 1992.8 1996.9 1992.7 10 10 10 10 10

Answer 1

1) Comparison of shapes

For the entire class (population) with n=526, the histogram shows normal distribution with skewed to left data points or say negatively skewed distributions of data points.

For sampling distribution with n=5, the histogram shows normal distribution with slightly skewed to left

For sampling distribution with n=10, the histogram shows normal distribution with kurtosis (kurtosis<0, mesokurtic) such that data values have high peak and light tails.

2) Comparison of means

The z-test is used to compare the population and sample mean

For population vs sample, n=5

The null and alternative hypotheses are,

$H_o:\mu_1=\mu_0$

$H_1:\mu_1\neq \mu_0$

The z-statistic is,

μ1-140 1999.4-1999 = 0.054 16 7t Vn

The critical value for significance level = 0.05 and a two-tailed test is,

$z_c=1.96$

Since,

2-0.054 <こcー1.96 Null hypothesis is failed to rejected

For population vs sample, n=10

The null and alternative hypotheses are,

$H_o:\mu_1=\mu_0$

$H_1:\mu_1\neq \mu_0$

The z-statistic is,

H1-o 1999.3- 1999 0.057 16 7t Vn ν'10

The critical value for significance level = 0.05 and a two-tailed test is,

$z_c=1.96$

Since,

z0.057 < 2c1.96 -> Null hypothesis is failed to rejected

3)

For population vs sample, n=5

The null and alternative hypotheses are,

$H_o:\sigma_1^2=\sigma^2$

$H_1:\sigma_1^2\neq \sigma^2$

The F-statistic is,

62.41-= F_στ 62.41 o? _ 275 56 _ 0.226

The critical value for significance level = 0.05 and a degree of freedom for numerator = 5 - 1 = 4 and degree of freedom for denominator = 526 - 1 = 525 is,

Since,

$F_L=0.121<F=0.226<F_U=2.81\Rightarrow \text{Null hypothesis is failed to rejected}$

For population vs sample, n=10

The null and alternative hypotheses are,

$H_o:\sigma_1^2=\sigma^2$

$H_1:\sigma_1^2\neq \sigma^2$

The F-statistic is,

$F=\frac{\sigma_1^2}{\sigma^2}=\frac{36}{275.56}=0.131$

The critical value for significance level = 0.05 and a degree of freedom for numerator = 10 - 1 = 9 and degree of freedom for denominator = 526 - 1 = 525 is,

F. = 0.299 and FU = 2.138

Since,

$F=0.131<F_L=0.299\Rightarrow \text{Null hypothesis is rejected}$

4)

a.

	Trial 1	Trial 2	Trial 3	Trial 4	Trial 5
	1974	2018	1986	1982	2018
	1982	2014	2014	1982	1986
	2014	2006	2006	1978	2000
	2014	1985	1985	1961	2006
	1986	2000	2014	2006	2017
mean	1994	2004.6	2001	1981.8	2005.4
z	-0.71393	0.713925	0.228995	-2.3573	0.821688
zc	1.644854	1.644854	1.644854	1.644854	1.644854
|z|>zc	No	No	No	Yes	No
				Unusual

b.

	Trial 1	Trial 2	Trial 3	Trial 4	Trial 5
mean	2000.1	1998.8	1992.8	1996.9	1992.7
z	0.152399	-0.09525	-1.23824	-0.4572	-1.25729
zc	1.644854	1.644854	1.644854	1.644854	1.644854
|z|>zc	No	No	No	No	No

No unusual at 5% significance level