Mathematical tools for natural sciences

This test is applied to the case when we sample data points from a population which follows a *normal distribution*, or *near normal distribution* for which the central limit theorem can be employed.

The n data points \(\small{x_1,x_2,.....,x_n} \) are assumed to be the random samples from a Gaussian (or near Gaussian) distribution of mean \(\small{\mu}\) and a *known* standard deviation \(\small{\sigma}\).

According to the central limit theorem, the Z statistic computed from the mean \(\small{\overline{x}}\) of n random samples follow a unit normal distribution given by,

\(~~~~~~~~~~~~~~~~~~~~~ \small{Z = \dfrac{\overline{x} - \mu}{\left(\dfrac{\sigma}{\sqrt{n}}\right)} = N(0,1) }\)

We proceed with the hypothesis testing as follows:

- We first compute the sample mean \(\small{\overline{x}}\) from the data.
- Knowing the value of mean \(\small{\mu}\) and the standard deviation \(\small{\sigma}\) of the population, we compute the value of Z using above expression.
- The statistical significance (also called "p-value") of this data is then obtained by computing the probability \(\small{P(\gt Z) }\) or \(\small{P(\lt -Z })\) from the unit normal distribution. Under the null hypothesis, the p-value represents the probability of getting the observed statistic Z.
- If the p-value is either smaller than a pre-decided value \(\small{\alpha}\) or the observed Z statistic is outside a given range (\(\small{-Z_0 \leq Z \leq Z_0) }\), we reject the null hypothesis and accept the alternate hypothesis.
- We can also reject the null hypothesis if the computed Z statistic for th data is outside the \(\small{(1-\alpha)100\%}\) confidence interval on the population mean.

Here, \(\small{Z_0}\) is the value of statistic above which the area under the unit normal curve is \(\small{\alpha}\).

For the given problem in hand, we an set our null hypothesis \(\small{H_0}\)and the alternate hypothesis \(\small{H_1}\) in one of the following three ways:

1. The population mean is equal to a particular value \(\small{\mu_0}\). A two sided hypothesis test.

\(~~~~~~~~~~~~~~~~~~~\small{H_0 : \mu = \mu_0} \)

\(~~~~~~~~~~~~~~~~~~~\small{H_A : \mu \neq \mu_0} \)

2. The population mean is greater than or equal to a particular value \(\small{\mu_0}\). A one sided hypothesis test.

\(~~~~~~~~~~~~~~~~~~~\small{H_0 : \mu \geq \mu_0} \)

\(~~~~~~~~~~~~~~~~~~~\small{H_A : \mu \lt \mu_0} \)

3. The population mean less than or equal to a particular value \(\small{\mu_0}\). A one sided hypothesis test.

\(~~~~~~~~~~~~~~~~~~~\small{H_0 : \mu \leq \mu_0} \)

\(~~~~~~~~~~~~~~~~~~~\small{H_A : \mu \gt \mu_0} \)

The female blood pressure of a certain population is known to follow Gaussian distribution with a mean 124.6 ad standard deviation 14.5 measured in units of mmHg. In order to test the effect of a food product on the female blood pressure, a clinical trial was performed in which 12 female volunteers of this population consumed the product for 3 months and their blood pressure were measured in the end. The readings are as follows:

141.5, 152.3, 121.2, 123.0, 151.6, 124.8, 138.9, 137.4, 145.6, 135.6, 135.4, 121.5

From this data, can we conclude that the population mean of the data set from which these random obserations are drawn is not equal to (ie., different from) 124.6? Let \(\small{\alpha = 0.05 }\) be the probability of rejecting the null hypothesis.

Since we want to test whether the population mean is not equal to 124.6,we set up the null and alternate hypothesis as follows: Since the alternate hypothesis can be satisfied by the values greater than or less than the given \(\small{\mu}\) value, this is a two sided test.

We first compute the sample mean of the data set. We get,

\(~~~~~~~~~\small{\overline{x} = 135.7 }\)

Under the null hypothesis, the sample mean should follow the central limit theorem given by,

\(~~~~~~~~~~~~~~~~~~~~~ \small{Z = \dfrac{\overline{x} - \mu}{\left(\dfrac{\sigma}{\sqrt{n}}\right)} = N(0,1) }\)

We compute this test statisic with \(\small{\mu=124.6,~\overline{x}=135.7,~~\sigma=14.5~~n=12}\):

\(~~~~~~~~~\small{Z = \dfrac{135.7 - 124.6 }{\left(\dfrac{14.5}{\sqrt{12}}\right)} =2.65 }\)

We have taken \(\small{\alpha =0.05}\) to be the probability for rejecting the null hypothesis. Since the rejection can occur due to sufficiently small as well as large values of the test statistic, the rejection probability \(\small{\alpha =0.05}\) is divided equally between these two areas to give \(\small{\alpha/2 =0.025}\).

What is the Z value for which the area above Z or area below -Z under the unit Gaussian is equal to \(\small{\alpha/2 =0.025}\)?. From the Gaussian table, we read this to be approximately 1.96.

We reject null hypothesis if the computed test statistic Z is either greater than 1.96 or less than -1.96. See the figure below:

**In our case, since the computed Z value of 2.65 is in the rejection region, the two sided null hypothesis is rejected to a significance level of 0.05. **

If the null hypothesis is true, what is the probability of getting the computed Z statistic?. This probability is called the"p-value" of the observed test statistic.

For the computed Z value of 2.65, the p-value is obtained from the Gaussian table to be \(\small{p = 0.004 }\). This is the area under the curve to the right of \(\small{Z = 2.65 }\) or to the left of \(\small{Z =-2.65 }\).

In general, for a 2 sided test, we reject the null hypothesis if \(\small{p \leq \alpha/2 }\).

If \(\small{p \gt \alpha/2 }\), we do not reject the null hypothesis.

For a significance level \(\small{\alpha = 0.05}\), the $95\%$ two sided confidence interval(CI) for the population mean is given by,

\(~~~~~~~~~~~~\small{CI~=~\overline{x} \pm Z_{0.975} {\dfrac{\sigma}{\sqrt{n}} }}\).

Substituting \(\small{\overline{x}=135.7,~~\sigma=14.5~~n=12}\) from the data and \(~\small{Z_{0.975}~=~1.96}~\) from Gaussian table, we get a $95\%$ confidence interval of \(\small{CI = 135.7 \pm 1.96*\dfrac{14.5}{\sqrt{12}}= 135.5 \pm 8.20 = (127.5, 143.9)}\)

In the Example-1 above, we tested whether the mean female blood pressure of the population is

Suppose, we want to test whether the mean female blood pressure of the population is

\(~~~~~~~~~~~\small{H_0 : \mu \leq 124.6 } \)

\(~~~~~~~~~~~\small{H_A : \mu \gt 124.6}\)

Though the null hypothesis is true for the infinite number of values of \(\small{\mu \leq 124.6}\), it is tested at only one value \(\small{\mu = 124.6 }\). If it is rejected at this value, it will be rejected at any value less than 124.6.

From Example-1, the computed value of statistics is \(\small{Z = 2.65 }\) and given \(\small{\alpha = 0.05 }\). Since this is a one sided test, the rejection region lies to the right of \(\small{Z_{1-\alpha} = Z_{0.95 } \approx 1.645 }\). Since the computed Z value of 2.65 lies in the rejection region, we reject the null hypothesis to conclude that the mean female blood pressure of the population is greater than 124.6. The alternate hypothesis is accepted. The rejection regions are marked below:

Alternately, from Example-1, the p-value corresponding to Z = 2.65 was obtained as 0.0040. Since this is less than \(\small{\alpha = 0.05 }\) for a one sided test, we can reject the null hypothesis to the significant level of 0.05.

For a significance level \(\small{\alpha = 0.05}\), the $95\%$ one sided confidence interval(CI) for the population mean is given by,

\(~~~~~~~~~~~~\small{CI~=~\overline{x} \pm Z_{0.95} {\dfrac{\sigma}{\sqrt{n}} }}\).

Substituting \(\small{\overline{x}=135.7,~~\sigma=14.5~~n=12}\) from the data and \(~\small{Z_{0.95}~=~1.644}~\) from Gaussian table, we get a $95\%$ confidence interval of \(\small{CI = 135.7 + 1.644*\dfrac{14.5}{\sqrt{12}}= 135.5+6.88 = (135.5, 142.6)}\)

The R script given below performs the one sample Z test. Given a data set x that is assumed to be randomly drawn from a Gaussian distribution of population mean mu and standard deviation sigma, the function returns the conclusions of the test along with computed statistic values.

The function is defined as,

one_sample_Z_test(x, sigma, muzero, alpha, null) wherex = data vectorsigma = population standard deviationmuzero = population mean for comparisonalpha = significane levelnull = string value indicating type of null hypothesis. Possible values of variable null are:"equal", "less_than_or_equal", "more_than_or_equal" The function returns a vector with two numbers :(p value, Z statistics) .

################################################### ## One sample Z test ## x = vector of data samples, which are numbers ## sigma = population standard deviation ## muzero = population mean for comparison ## alpha = significance level for testing ## null = string with three possible values "equal, "greater_than_or_equal,"less_than_or_equal" for indicating whether the test is one sided or two sided. one_sample_Z_test = function(x, sigma, muzero, alpha, null ){ ## compute sample mean xbar = mean(x) ## get the sample size n = length(x) ## compute the Z statistic Z_statistic = (xbar - muzero)/(sigma/sqrt(n)) ## compute the p-value pvalue = 1.0 if(Z_statistic > 0) pvalue = 1 - pnorm(Z_statistic) if(Z_statistic & 0) pvalue = pnorm(Z_statistic) if(Z_statistic == 0) pvalue = 0.5 ## Perform the statitical test by comaring the computed Z statistic with the ## critical value for various cases ### Case 1 : Null hypothesis that populatin mean equals a given value if(null == "equal") { Z_critical = qnorm(1 - (alpha/2)) print("################################################################") print("One sample Z test : ") print(paste("sample size = ", n)) if( (Z_statistic > Z_critical) | (Z_statistic < -Z_critical) ) { print("One sample Z test : ") print(paste("sample size = ", n)) print(paste("Null hypothesis is rejected at the level of significance ", alpha/2)) print(paste("Population mean not equal to ", muzero)) print(paste("p value for the test = ", round(pvalue, digits=5))) print(paste("Value of Z statistic = ", round(Z_statistic, digits=2))) print(paste("Critical value of the test = ", round(Z_critical, digits=2))) resultVec = c(round(pvalue, digits=5), round(Z_statistic, digits=2)) } if( (Z_statistic < Z_critical) & (Z_statistic > -Z_critical) ) { print("One sample Z test : ") print(paste("sample size = ", n)) print(paste("Null hypothesis is accepted at the level of significance ", alpha/2)) print(paste("Population mean equal to ", muzero)) print(paste("p value for the test = ", round(pvalue, digits=5))) print(paste("Value of Z statistic = ", round(Z_statistic, digits=2))) print(paste("Critical value of the test = ", round(Z_critical, digits=2))) resultVec = c(round(pvalue, digits=5), round(Z_statistic, digits=2)) } } ##### Case 2 : Null hypothesis that population mean is less than or equal to a given value if(null == "less_than_or_equal") { Z_critical = qnorm(1 - alpha) print("################################################################") print("One sample Z test : ") print(paste("sample size = ", n)) if( Z_statistic > Z_critical ) { print("One sample Z test : ") print(paste("sample size = ", n)) print(paste("Null hypothesis is rejected at the level of significance ", alpha)) print(paste("Population mean greater than ", muzero)) print(paste("p value for the test = ", round(pvalue, digits=5))) print(paste("Value of Z statistic = ", round(Z_statistic, digits=2))) print(paste("Critical value of the test = ", round(Z_critical, digits=2))) resultVec = c(round(pvalue, digits=5), round(Z_statistic, digits=2)) } if( Z_statistic <= Z_critical ) { print("One sample Z test : ") print(paste("sample size = ", n)) print(paste("Null hypothesis is accepted at the level of significance ", alpha)) print(paste("p value for the test = ", round(pvalue, digits=5))) print(paste("Value of Z statistic = ", round(Z_statistic, digits=2))) print(paste("Critical value of the test = ", round(Z_critical, digits=2))) resultVec = c(round(pvalue, digits=5), round(Z_statistic, digits=2)) } } ###### Case 3 : Null hypothesis that the population mean is less than or equal to a given value. if(null == "greater_than_or_equal") { Z_critical = qnorm(1 - alpha) print("################################################################") print("One sample Z test : ") print(paste("sample size = ", n)) if( Z_statistic &l Z_critical ) { print("One sample Z test : ") print(paste("sample size = ", n)) print(paste("Null hypothesis is rejected at the level of significance ", alpha)) print(paste("Population mean is less than ", muzero)) print(paste("p value for the test = ", round(pvalue, digits=5))) print(paste("Value of Z statistic = ", round(Z_statistic, digits=2))) print(paste("Critical value of the test = ", round(Z_critical, digits=2))) resultVec = c(round(pvalue, digits=5), round(Z_statistic, digits=2)) } if( Z_statistic >= Z_critical ) { print("One sample Z test : ") print(paste("sample size = ", n)) print(paste("Null hypothesis is accepted at the level of significance ", alpha)) print(paste("p value for the test = ", round(pvalue, digits=5))) print(paste("Value of Z statistic = ", round(Z_statistic, digits=2))) print(paste("Critical value of the test = ", round(Z_critical, digits=2))) resultVec = c(round(pvalue, digits=5), round(Z_statistic, digits=2)) } } return(resultVec) } ## end of the function ###############------------------------------------------------ ## Perform a sample test with the function ## define a data set x = c(141.5, 152.3, 121.2, 123.0, 151.6, 124.8, 138.9, 137.4, 145.6, 135.6, 135.4, 121.5) ## mean to be compared muzero = 124.6 #population standard deviation sigma = 14.5 ## alpha value alpha = 0.05 ## call the function. "res" is a vector with p-vlue and Z value for the test. res = one_sample_Z_test(x, sigma, muzero, alpha, "equal") print(res)

Executing the above script in R prints the following results and figures of probability distribution on the screen:

[1] "################################################################" [1] "One sample Z test : " [1] "sample size = 12" [1] "One sample Z test : " [1] "sample size = 12" [1] "Null hypothesis is rejected at the level of significance 0.025" [1] "Population mean not equal to 124.6" [1] "p value for the test = 0.00391" [1] "Value of Z statistic = 2.66" [1] "Critical value of the test = 1.96" [1] 0.00391 2.66000