# One sample t test for the mean - overview

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One sample $t$ test for the mean
One sample $z$ test for the mean
Independent variableIndependent variable
NoneNone
Dependent variableDependent variable
One quantitative of interval or ratio levelOne quantitative of interval or ratio level
Null hypothesisNull hypothesis
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis
Alternative hypothesisAlternative hypothesis
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
Two sided: $\mu \neq \mu_0$
Right sided: $\mu > \mu_0$
Left sided: $\mu < \mu_0$
AssumptionsAssumptions
• Scores are normally distributed in the population
• Sample is a simple random sample from the population. That is, observations are independent of one another
• Scores are normally distributed in the population
• Population standard deviation $\sigma$ is known
• Sample is a simple random sample from the population. That is, observations are independent of one another
Test statisticTest statistic
$t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $s$ is the sample standard deviation, $N$ is the sample size.

The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$
$z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $\sigma$ is the population standard deviation, $N$ is the sample size.

The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$
Sampling distribution of $t$ if H0 were trueSampling distribution of $z$ if H0 were true
$t$ distribution with $N - 1$ degrees of freedomStandard normal
Significant?Significant?
Two sided:
Right sided:
Left sided:
Two sided:
Right sided:
Left sided:
$C\%$ confidence interval for $\mu$$C\% confidence interval for \mu \bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}} where the critical value t^* is the value under the t_{N-1} distribution with the area C / 100 between -t^* and t^* (e.g. t^* = 2.086 for a 95% confidence interval when df = 20) The confidence interval for \mu can also be used as significance test. \bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}} where z^* is the value under the normal curve with the area C / 100 between -z^* and z^* (e.g. z^* = 1.96 for a 95% confidence interval) The confidence interval for \mu can also be used as significance test. Effect sizeEffect size Cohen's d: Standardized difference between the sample mean and \mu_0:$$d = \frac{\bar{y} - \mu_0}{s}$$Indicates how many standard deviations s the sample mean \bar{y} is removed from \mu_0 Cohen's d: Standardized difference between the sample mean and \mu_0:$$d = \frac{\bar{y} - \mu_0}{\sigma}$$Indicates how many standard deviations$\sigma$the sample mean$\bar{y}$is removed from$\mu_0$Visual representationVisual representation Example contextExample context Is the average mental health score of office workers different from$\mu_0$= 50?Is the average mental health score of office workers different from$\mu_0$= 50? Assume that the standard deviation of the mental health scores in the population is$\sigma$= 3. SPSSn.a. Analyze > Compare Means > One-Sample T Test... • Put your variable in the box below Test Variable(s) • Fill in the value for$\mu_0$in the box next to Test Value - Jamovin.a. T-Tests > One Sample T-Test • Put your variable in the box below Dependent Variables • Under Hypothesis, fill in the value for$\mu_0\$ in the box next to Test Value, and select your alternative hypothesis
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Practice questionsPractice questions