Lecture 14: Hypothesis Test for One Variance

STAT 205: Introduction to Mathematical Statistics

Dr. Irene Vrbik

University of British Columbia Okanagan

March 17, 2024

Introduction

We have covered three hypothesis tests for a single sample:

1. Hypothesis test for the mean \(\mu\) with \(\sigma\) known (\(Z\)- test)
2. Hypothesis tests for the proportion \(p\) (\(Z\)- test)
3. Hypothesis test for the mean \(\mu\) with \(\sigma\) unknown (\(t\)-test)

Today we consider hypothesis tests involve the population variance \(\sigma^2\)

Assumptions: \(X_1, X_2, \dots, X_n\) are i.i.d + assumptions in the rhombuses.

Recap

In Lecture 7 we saw how to construct a confidence interval for \(\sigma^2\) based on the sampling distribution derived in Lecture 8.

For random samples from normal populations, we know:

\[ \dfrac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1} \]

where \(S^2 = \frac{\sum_{i = 1}^n (X_i - \bar{X})}{n-1}\) is the sample variance and \(\chi^2_{n-1}\) is the Chi-squared distribution with \(n-1\) degrees of freedom.

Hypotheses

• We may which to test if there is evidence to suggest that population variance differs for some hypothesized value \(\sigma_0^2\).

• As before, we start with a null hypothesis (\(H_0\)) that the population variance equals a specified value (\(\sigma^2 = \sigma_0^2\))

• We test this against the alternative hypothesis \(H_A\) which can either be one-sided (\(\sigma^2 < \sigma_0^2\) or \(\sigma^2 > \sigma_0^2\)) or two-sided (\(\sigma^2 \neq \sigma_0^2\)).

• this could be important whenever it is desired to control the uniformity of a product or an operation.

• eg. Heinz, even if we’re getting 20oz average, we don’t want the variance to be out of control

Test Statistic

Recall that our test statistic is calculated assuming the null hypothesis is true. Hence, if we are testing \(H_0: \sigma^2 = \sigma_0^2\), the test statistic we use is : \[ \chi^2 = \dfrac{(n-1)S^2}{\sigma_0^2} \] where \(\chi^2 \sim \chi^2_{n-1}\).

So \(H_0\) is true, and the population is normal, this test statistic will have a chi-square dist with \(n-1\) degrees for freedom

Chi-square distrbituion

Assumptions

For the following inference procedures to be valid we require:

1. A simple random sample from the population
2. A normally distributed population (very important, even for large sample sizes)

Warning

It is important to note that if the population is not approximately normally distributed, chi-squared distribution may not accurately represent the sampling distribution of the test statistic.

• Inference procedures are quite robust to the violation of normality (meaning they perform well for many violations of the normalility assumption)

• inference procedures for the variance are heavily reliant on the nomaility assumption.

• that the procedures discussed in this lecture can perform very poorly when the normality assumption is violated. a bit dubious

• Important:

Rejection Regions and \(p\)-values for the chi-square test concerning one variance

Alternative	Reject \(H_A\) if…	\(p\)-value
\(H_A: \sigma^2 < \sigma_0^2\)	\(\chi^2_{\text{obs}} \geq \chi^2_\alpha\)	Area to the right of \(\chi^2_{\text{obs}}\)
\(H_A: \sigma^2 > \sigma_0^2\)	\(\chi^2_{\text{obs}} \leq \chi^2_{1-\alpha}\)	Area to the left of \(\chi^2_{\text{obs}}\)
\(H_A: \sigma^2 \neq \sigma_0^2\)	\(\chi^2_{\text{obs}} \geq \chi^2_{\alpha/2}\) or \(\chi^2_{\text{obs}} \leq \chi^2_{1-\alpha/2}\)	Double the area to the left or right of \(\chi^2_{\text{obs}}\); whichever is smallest.

Using similar concepts to those discussed in previous lecture, we will see how we can conduct hypothesis tests involving \(\sigma^2\).

Critical Region (upper-tailed)

The rejection region associated with an upper-tailed test for the population variance. Note that the critical value will depend on the chosen significance level (\(\alpha\)) and the d.f.

Critical Region (lower-tailed)

The rejection region associated with an upper-tailed test for the population variance. Note that the critical value will depend on the chosen significance level (\(\alpha\)) and the d.f.

Critical Region (two-tailed)

The rejection region associated with an upper-tailed test for the population variance. Note that the critical value will depend on the chosen significance level (\(\alpha\)) and the d.f.

P-values

Similarly we can find \(p\)-values from Chi-squared tables or R

\(p\)-value for lower-tailed: \[\Pr(\chi^2 < \chi^2_{\text{obs}})\] \(p\)-value for upper-tailed: \[\Pr(\chi^2 > \chi^2_{\text{obs}})\] \(p\)-value for two-tailed:

\[2\cdot \min \{ \Pr(\chi^2 < \chi^2_{\text{obs}}), \Pr(\chi^2 > \chi^2_{\text{obs}})\}\]

P-values

Similarly we can find \(p\)-values from Chi-squared tables or R

\(p\)-value for lower-tailed: \[\Pr(\chi^2 < \chi^2_{\text{obs}})\] \(p\)-value for upper-tailed: \[\Pr(\chi^2 > \chi^2_{\text{obs}})\] \(p\)-value for two-tailed:

\[2\cdot \min \{ \Pr(\chi^2 < \chi^2_{\text{obs}}), \Pr(\chi^2 > \chi^2_{\text{obs}})\}\]

Double the area to the left or right of \(\chi^2_{\text{obs}}\); whichever is smallest.

Exercise 1: Beyond Burger Fat

Beyond Burgers claim to have 18g grams of fat. A random sample of 6 burgers had a mean of 19.45 and a variance of 0.85 grams\(^2\). Suppose that the quality assurance team at the company will on accept at most a \(\sigma\) of 0.5. Use the 0.05 level of significance to test the null hypotehsis \(\sigma = 0.5\) against the appropriate alternative.

\[\begin{align} H_0: \sigma^2 &= 0.5^2 & H_A: \sigma^2 &> 0.5^2 \end{align}\]

Distribution of Test Statistic

par(mar=c(4,4,0,0) + 0.1)
curve(dchisq(x, df = n-1), from = 0, to = 20, ylab = "Density", xlab = expression(chi^2))

Under the null hypothesis, the test statistic follows \(\chi^2 = (n-1)S^2/0.5^2\) a chi-square distribution with df = 5

If this alternative hypothesis is true and \(\sigma^2\) squared is greater than the hypothesized value then the sample variance \(S^2\) will have a tendency to be greater than the hypothesized value and the test statistic will tend to be large

• here large values of the test statistic give evidence against the null hypothesis and in favor of this alternative hypothesis

• the p-value is the probability under the null hypothesis of getting the observed value of the test statistic or something even greater and that’s simply this area the area to the right of the observed value of the test statistic under the chi-square distribution with n minus 1

Critical value

The critical value can be found by determining what value on the chi-square curve with 5 df yield a 5 percent probability in the upper tail (since we are doing an upper-tailed test). In R: qchisq(alpha, df=n-1, lower.tail = FALSE). Verify using \(\chi^2\) table.

Observed Test Statistic

Compute the observed test statistic which we denote by \(\chi^2_{\text{obs}}\)

\[\begin{align} \chi^2_{\text{obs}} &= \dfrac{(n-1)s^2}{\sigma_0^2}\\ &= \dfrac{(5)0.85^2}{0.5^2}\\ &= \dfrac{3.6125}{0.25} = 14.45 \end{align}\]

Critical value

Since the observed test statistic falls in the rejection region, i.e. \(\chi^2_{\text{obs}} > \chi^2_{\alpha}\), we rejection the null hypothesis in favour of the alternative.

P-value in R

n = 6; s = 0.85; sig0 = 0.5
x_obs = ((n-1)*s^2)/(sig0^2)
pval = pchisq(x_obs, df = n-1, lower.tail = FALSE)

Alternatively we could compute the p-value which in this case is 0.013. Since this is smaller than the alpha-level of 0.05, we reject the null hypothesis in favour of the alternative. Verify using \(\chi^2\) table.

P-value from tables

Using the chi-square distribution table we can see that our observed test statistic falls between two values. We can use the neigbouring values to approximate our p-value.

Approximate P-value

It is clear from the visualization that \[\begin{align} \Pr(\chi^2_{5} > \chi^2_{0.025}) > \Pr(\chi^2_{5} > \chi^2_{\text{obs}})\\ \Pr(\chi^2_{5} > \chi^2_{\text{obs}}) < \Pr(\chi^2_{5} > \chi^2_{0.01}) \\ \end{align}\]

The \(p\)-value, \(\Pr(\chi^2_{5} > 14.45)\) can then be expressed as: \[\begin{align} 0.01 < p\text{-value } < 0.025 \end{align}\]

Conclusion

Since:

1. the \(p\)-value (0.013) is less than \(\alpha\) = 0.05 OR
2. the the observed test statistic (\(\chi^2_{\text{obs}}\) = 14.45) is larger than the critical value \(\chi^2_{\alpha}\)

we reject the null hypothesis in favour of the alternative. More specifically, there is very strong evidence to suggest that the population variance \(\sigma^2\) is greater than \(0.5^2\).