Six soil samples taken from a particular region were subjected to chemical analysis to determine the pH of each sample, yielding a sample mean pH of 5.9 and a sample standard deviation of .62. Assume the distribution of soil pH in this region is normal.
Compute the upper limit of a 95% confidence interval for the true mean pH of soil in this region.
Solution
Sample mean
The sample mean of $X$ is $\overline{x}=5.9\text{ }$.
Sample standard deviation
The sample standard deviation is $s=0.62 \text{ }$
Sample size $n = 6$, sample mean $\overline{X}= 5.9$, sample standard deviation $s = 0.62$.
The confidence level is $1-\alpha = 0.95$.
Step 1 Specify the confidence level $(1-\alpha)$
Confidence level is $1-\alpha = 0.95$. Thus, the level of significance is $\alpha = 0.05$.
Step 2 Given information
Sample size $n =6$, sample mean $\overline{X}=5.9$, sample standard deviation $s=0.62$.
Step 3 Specify the formula
$100(1-\alpha)$% confidence interval for the population mean $\mu$ is
$$ \begin{aligned} \overline{X} - E \leq \mu \leq \overline{X} + E \end{aligned} $$
where $E = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}$, andand $t_{\alpha/2, n-1}$ is the $t$ value providing an area of $\alpha/2$ in the upper tail of the students' $t$ distribution.
Step 4 Determine the critical value
The critical value of $t$ for given level of significance and $n-1$ degrees of freedom is $t_{\alpha/2,n-1}$.
Thus $t_{\alpha/2,n-1} = t_{0.025,6-1}= 2.571$.
Step 5 Compute the margin of error
The margin of error for mean is
$$ \begin{aligned} E & = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}\\ & = 2.571 \frac{0.62}{\sqrt{6}} \\ & = 0.6508. \end{aligned} $$
Step 6 Determine the confidence interval
$95$% confidence interval estimate for population mean is
$$ \begin{aligned} \overline{X} - E & \leq \mu \leq \overline{X} + E\\ 5.9 - 0.651 & \leq \mu \leq 5.9 + 0.651\\ 5.2492 &\leq \mu \leq 6.5508. \end{aligned} $$
Thus, $95$% confidence interval estimate for population mean is $(5.2492,6.5508)$.
Interpretation
We can be $95$% confident that the true mean pH of soil in this region would have limits $5.2492$ and $6.5508$.
Thus the upper limit of $95$% confidence interval for the true mean pH of soil in this region is $6.5508$.
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
