11.2: One Mean T Test
- Page ID
- 105861
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Next, we will learn how to apply the One Mean \(T\) Procedure to test a statistical claim about a population mean. Consider the following example.
An incubation period is a time between when you contract a virus and when your symptoms start. Assume that the population of incubation periods for a novel coronavirus is normally distributed. By surveying randomly selected local hospitals, a researcher was able to obtain the following sample of incubation periods of \(10\) patients:
6, 2, 4, 3, 3, 7, 5, 8, 5, 7
At \(5\%\) significance level, test the claim that the average incubation period of the novel coronavirus is less than \(6\) days.
Note that the average of the sample is \(\bar{x}_i=5\) and since the population standard deviation is not given, we compute the sample standard deviation instead \(s=2\).
Now, let’s identify the statistical claim that needs to be tested:
“the average incubation period of the novel coronavirus is less than 6 days”
The key word “average” suggests that the claim is about the parameter \(\mu\), so we can symbolically express the claim as
\(\mu<6\)
Since the claim is about the population mean and the standard deviation is unknown, we will use the one mean \(T\) procedure. Let’s check if all necessary assumptions are satisfied:
- The sample is simple random
- The population is normally distributed
- The population standard deviation, \(\sigma\), is unknown but \(s=2\)
We will use the following template to perform the hypothesis testing:
In step 1, we will set up the hypothesis:
Since the claim is in the form of an inequality, therefore it must be set as an alternative hypothesis, therefore the null hypothesis is \(\mu=6\) and the test is left-tail, that is:
|
\(H_0: \mu=6\) \(H_a: \mu<6\)
|
LT |
In step 2, we will identify the significance level:
The significance level can always be found in the text of the problem. In our case it is \(5\%\), thus:
\(\alpha=0.05\)
In step 3, we will find the test statistic using the formula:
\(t_0=\frac{\bar{x}_i-\mu_0}{s/\sqrt{n}}=\frac{5-6}{2/\sqrt{10}}=-1.58\)
In step 4, we will perform either the critical value approach or p-value approach to test the claim:
- In critical value approach, we construct the rejection region using the \(T\)-curve with \(df=n-1=10-1=9\):
RR: less than \(-t_{0.05}=-1.833\)
- In p-value approach, we compute the p-value using the \(T\)-curve with \(df=n-1=10-1=9\):
P-Value: \(P(T<-1.58)=0.074\)
In step 5, we will draw the conclusion:
- In the critical value approach, we must check whether the test statistic is in the rejection region or not. Our test statistic is \(-1.58\) and it is to the right of the critical value \(-1.833\), thus it is not in the rejection region.
- In the p-value approach, we must check whether the p-value is less than the significance level or not. Our p-value is \(0.074\) and it is greater than \(\alpha\) .
Both tests suggest that we DO NOT reject the null hypothesis in favor of the alternative.
In step 6, we will interpret the results:
Under \(5\%\) significance level we DO NOT have sufficient evidence to suggest that the mean incubation period is less than \(6\) days.
We discussed how to apply the One Mean T Procedure to test a statistical claim about a population mean when the population standard deviation is not given.