One Sample Hypothesis Test (t-test)

Introduction: One Sample Hypothesis Test

In this post, I will discuss One Sample Hypothesis Test (One Sample t-test). When testing a claim about the mean using sample data with a small number of observations (i.e., sample size), the appropriate t-distribution instead of the standard normal distribution should be used to determine the standardized test statistic, critical values, rejection region, and p-values.

Recall that if the sample of values drawn follows the normal distribution, the sample size (number of observations in the sample) is less than 30, and the population standard deviation is unknown, then the random variable

$$t=\frac{\overline{x} – \mu}{\frac{s}{\sqrt{n}}}$$

has the Student’s t-distribution with $n-1$ degrees of freedom.

The procedure of locating the rejection regions for a t-distribution hypotheses test is the same as for the normal distribution tests, however, the critical values will differ. To find the critical value(s) $t_0$ for a test, determine if the test is one-tailed or two-tailed and the significance level ($\alpha$). The critical values can be found in the t-distribution table by looking up the entry in the column giving the level of significance and the row showing the degrees of freedom.

Note that:

  • For a right-tailed test, $t_0$ is the positive value in the table
  • For a left-tailed test, $t_0$ is the negative of the value in the table
  • For a two-tailed test, there are two critical values $t_0$ both the value and its opposite.
One Sample Hypothesis Testing-Tails-Critical-Region

Assumptions of the One Sample Hypothesis Test (t-test)

  • Independence: Observations in the sample should be independent of each other.
  • Normality: The population from which the sample is drawn should be normally distributed. However, the t-test is relatively robust to violations of normality, especially for larger sample sizes.
  • Random Sampling: The sample should be a random sample from the population.

One Sample Hypothesis Test for Mean

Example 1: SAT Math scores are normally distributed. A sample of SAT Math scores for 16 students has an average score of 522.8 with a sample standard deviation of 154.5. Suppose, one wishes to support the claim that the average SAT Math score exceeds 500 using a level of significance of 0.05.

Solution

Step 1: The null and alternative hypotheses test in this case are

$H_0: \mu \le 500$ vs $H_1: \mu > 500$

From the alternative hypothesis, the test is right-hand-tailed with $\mu_0=500$.

Step 2: Level of Significance is 5% = 0.05

Step 3: Critical Value

Using the t-distribution table with one Tail, 5% level of significance, and $n-1=16-1=15$ degrees of freedom, the critical value is $t_0=1.753$. The rejection region is thus $t\ge 1.7535$.

Step 4: Test Statistics

The standardized test statistic is

\begin{align*}
t&=\frac{\overline{X} – \mu_0 }{\frac{s}{\sqrt{n}}}\\
&= \frac{522.8 – 500}{\frac{154.5}{\sqrt{16}}}\\
&= \frac{22.8}{38.625} = 0.59
\end{align*}

Step 5: Interpretation of One Sample Mean Test

Since the standardized test statistic is not in the region of rejection, therefore, one should not reject $H_0$ and so the sample data is not sufficient to support the claim that the average exceeds 500 at the 0.05 level of significance.

Example 2: A biologist measures the weights of anesthetized female grizzly bears during winter. A sample of 14 bears is found to have an average weight of $\overline{X} = 376.6$lbs. with a sample standard deviation of $s=32.5$lbs. Is there sufficient evidence to support the claim that the average weight of all female bears in the area is less than 400 lbs? Use $\alpha=0.01$ level of significance.

Solution:

Step 1: The null and alternative hypotheses in this case

$H_0:\mu \ge 400$ vs $H_1:\mu < 400$

From the alternative hypothesis, the test is left-hand-tailed with $\mu_0=400$.

Step 2: Level of Significance is 1% = 0.01

Step 3: Critical Value

Using the Student’s t-distribution table with one tail, the level of significance $\alpha = 0.01$, and $n-1=14-1=13$ degrees of freedom, the critical value $t_0=-2.650$. The region of rejection is thus $t\le -2.650$.

Step 4: The Test Statistics is

\begin{align*}
t&=\frac{\overline{X} – \mu_0 }{\frac{s}{\sqrt{n}}}\\
&= \frac{376.6 – 400}{\frac{32.5}{\sqrt{14}}}\\
&= \frac{-23.4}{8.686} = -2.694
\end{align*}

Since the standardized test statistic is in the rejection region, one should reject the null hypothesis ($H_0$), which supports the claim that the average bear weight is less than 400 lbs.

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Critical Values and Rejection Region

In statistical hypotheses testing procedure, an important step is to determine whether to reject the null hypothesis. The step is to compute/find the critical values and rejection region.

Rejection Region and Critical Values

A rejection region for a hypothesis test is the range of values for the standardized test statistic which would lead us to decide whether to reject the null hypothesis. The Critical values for a hypothesis test are the z-scores which separate the rejection region(s) from the non-rejection region (also called the acceptance region of $H_0$).  The critical values will be denoted by $Z_0$.

The rejection region for a test is determined by the type of test (left-tailed, right-tailed, or two-tailed) and the level of significance (denoted by $\alpha$) for the test. For a left-tailed test, the rejection region is a region in the left tail of the normal distribution, for a right-tailed test, it is in the right tail, and for a two-tailed test, there are two equal rejection regions in either tail.

Hypothesis-Testing-Tails-Critical Values and Rejection Region

Once we establish the critical values and rejection region, if the standardized test statistics for a sample data set fall in the region of rejection, the null hypothesis is rejected.

Examples: Critical Values and Rejection Region

Example 1: A university claims that the average SAT score for its incoming freshmen is 1080. A sample of 56 freshmen at the university is drawn and the average SAT score is found to be $\overline{x}=1044$ with a sample standard deviation of $s=94.7$ points.

    In the above SAT example, the test is two-tailed, so the rejection region will be the two tails at either end of the normal distribution. If we again want $\alpha=0.05$, then the area under the curve in both rejection regions together should be 0.05. For this purpose, we will look up $\frac{\alpha}{2}=0.025$ in the standard normal table to get critical values of $Z_0 = \pm 1.96$. The rejection region thus consists of $Z \le 1.96$ and $Z\ge 1.96$. Since the standardized test statistic $Z=-2.85$ falls in the region, the university’s claim of $\mu = 1080$ would be rejected in this case.

    Example 2: Consider a left-tailed Z test. For a 0.05 level of significance, the rejection region would be the values in the lowest 5% of the standard normal distribution (5% lowest area under the normal curve). In this case, the critical value (the corresponding) Z-score will be $-1.645$. So the critical value $Z_0$ will be $-1.645$ and the rejection region will be $Z\le -1.645$.

    Note that for the case of right-tailed the rejection region would be the values in the highest 5% of the standard normal distribution table. The Z-score will be $1.645$ and the rejection region will be $Z\ge 1.645$.

    Hypothesis Test

    Exercise: Critical Values and Rejection Region

    1. Find the critical values and rejection regions(s) for the standardized Z-test of the following:
    • A right-tailed test with $\alpha = 0.05$
    • A left-tailed test with $\alpha = 0.01$
    • A two-tailed test with $\alpha = 0.10$
    • A right-tailed test with $\alpha = 0.02$
    1. Mercury levels in fish are considered dangerous to people if they exceed 0.5mg mercury per kilogram of meat. A sample of 50 tuna is collected, and the mean level of mercury in these 50 fishes is 0.6m/kg, with a standard deviation of 0.2mg/kg. A health warning will be issued if the claim that the mean exceeds 0.5mg/kg can be supported at the $\alpha=0.10$ level of significance. Determine the null and alternative hypotheses in this case, the type of the test, the critical value(s), and the rejection region. Find the standardized test statistics for the information given in the exercise. Should the health warning be issued?

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    Important MCQs Hypothesis Statistics Quiz 6

    The post is about the MCQs Hypothesis Statistics Quiz. There are 20 multiple-choice questions covering topics related to hypothesis testing, level of significance, test statistics, critical region, and one-tailed and two-tailed tests. Let us start with MCQs Hypothesis Statistics Quiz.

    MCQs Hypothesis Testing

    1. Type-I error will occur if an innocent person is

     
     
     
     

    2. A deserving player is not selected for the national team, it is an example of

     
     
     
     

    3. Which of the following is (are) considered to be inferential statistics?

     
     
     
     

    4. A ________ error is made if $H_1$ is true but $H_0$ is accepted

     
     
     
     

    5. If $\alpha=0.05$%, the value of one-tailed $Z$ test will be

     
     
     
     

    6. The probability of rejecting a true hypothesis is called

     
     
     
     

    7. The probability of rejecting a false $H_0$ is

     
     
     
     

    8. Which of the following statistics can be used to determine whether or not there is a statistically significant relationship between two variables in a contingency table?

     
     
     
     

    9. In a Z-test, the number of degrees of freedom is

     
     
     
     

    10. A .05 level of significance means that

     
     
     
     

    11. The choice of a one-tailed test and a two-tailed test depends upon

     
     
     
     

    12. The region of acceptance of $H_0$ is called

     
     
     
     

    13. The level of significance is the risk of

     
     
     
     

    14. Which one is an example of a two-tailed test

     
     
     
     

    15. The region of rejection of $H_0$ is called

     
     
     
     

    16. A statistic on the basis of which a decision is made about the hypothesis of interest is called

     
     
     
     

    17. If population standard deviation is known and $n>30$ then appropriate test statistics mean comparison is

     
     
     
     

    18. When a critical region is located on both sides of the curve, it is called

     
     
     
     

    19. Fisher exact test is used for:

     
     
     
     

    20. In a Wilcoxon rank sum test

     
     
     
     

    MCQs Hypothesis Statistics Quiz

    MCQs Hypothesis Statistics Quiz
    • Type-I error will occur if an innocent person is
    • A deserving player is not selected for the national team, it is an example of
    • A ———- error is made if $H_1$ is true but $H_0$ is accepted
    • A statistic on the basis of which a decision is made about the hypothesis of interest is called
    • The probability of rejecting a true hypothesis is called
    • The probability of rejecting a false $H_0$ is
    • The level of significance is the risk of
    • When a critical region is located on both sides of the curve, it is called
    • The choice of a one-tailed test and a two-tailed test depends upon
    • Which one is an example of a two-tailed test
    • The region of acceptance of $H_0$ is called
    • The region of rejection of $H_0$ is called
    • In a Z-test, the number of degrees of freedom is
    • If $\alpha=0.05$%, the value of one-tailed $Z$ test will be
    • If population standard deviation is known and $n>30$ then appropriate test statistics mean comparison is
    • In a Wilcoxon rank sum test
    • Fisher exact test is used for:
    • Which of the following statistics can be used to determine whether or not there is a statistically significant relationship between two variables in a contingency table?
    • Which of the following is (are) considered to be inferential statistics?
    • A .05 level of significance means that
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    Important MCQs Hypothesis and Hypothesis Testing Quiz 5

    The post is about the Hypothesis and Hypothesis Testing Quiz. There are 20 multiple-choice questions covering topics related to the statement of null and alternative hypothesis, level of significance, test statistics, critical region, and parametric and non-parametric tests. Let us start with the MQs Hypothesis and Hypothesis Testing Quiz.

    Please go to Important MCQs Hypothesis and Hypothesis Testing Quiz 5 to view the test

    Hypothesis and Hypothesis Testing Quiz

    MCQs Hypothesis and Hypothesis Testing Quiz
    • The first and starting point in hypothesis testing is
    • A hypothesis which is to be tested for the possible nullification is called
    • The hypothesis, which is being tested is
    • Which of the following can be $H_1$
    • Which of the following can be an alternative hypothesis $H_1$
    • A hypothesis in which all parameters are specified is called
    • If $H_0:\mu =10$ and $\sigma = 5$, then it is
    • Which one of the following cannot be a null hypothesis
    • The probability of type-I error is denoted by
    • The probability of rejecting $H_0$, when $H_0$ is true is called
    • Rejecting $H_0$ when $H_0$ is false is called
    • Rejecting $H_0$ when $H_0$ is true is called
    • A judge can release(acquit) a guilty person is an example of
    • A misfit person is not selected for a job is
    • A good scheme related to education is rejected by the education department, it is an example of
    • The greater the value of the F-ratio
    • If a hypothesis test were conducted using $\alpha=0.05$, for which of the following p-values, the null hypothesis be rejected:
    • To test $H_0:\sigma^2 = \sigma_o^2$, we use:
    • Nonparametric tests are used when the level of measurement is
    • The nonparametric equivalent of an unpaired samples t-test is the
    Hypothesis and hypothesis Testing Quiz

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