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.