# Hypothesis testing concluded

05/03/2021

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## Outline

• The following topics will be covered in this lecture:
• General hypothesis testing of the mean

## General hypothesis testing of the mean

• When we studied confidence intervals, we differentiated how we could construct these in the case where the population variance was known or unknown.

• Particularly, we had to use the sample estimate for the standard error,

$\frac{s}{\sqrt{n}}$ in place of the true standard error in our calculations.

• Under the assumption that the random sample $$X_i$$ is normally distributed, we found that

$T = \frac{X - \overline{x}}{\frac{s}{\sqrt{n}}}$ is t distributed in $$n-1$$ degrees of freedom.

• Using the associated $$t_\frac{\alpha}{2}$$ critical value, we calculated the $$(1-\alpha)\times 100\%$$ confidence interval as

$\left( \overline{x} - \hat{\sigma}_\overline{X} t_\frac{\alpha}{2} , \overline{x} + \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}\right).$

• With the same development in the case $$\sigma^2$$ is known, we can replace the $$z_\frac{\alpha}{2}$$ and $$z_\alpha$$ critical values with t critical values to produce general hypothesis tests.

### General hypothesis testing of the mean

• Specifically, if we have a formal hypothesis test

\begin{align} H_0:\mu = \tilde{\mu} & & H_1: \mu \neq \tilde{\mu}; \end{align} and if the variance $$\sigma^2$$ is also unknown;

• then assuming the null, we write the acceptance region as

$\left( \tilde{\mu} - \hat{\sigma}_\overline{X} t_\frac{\alpha}{2} , \tilde{\mu} + \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}\right).$

• If the sample mean $$\overline{X}$$ lies outside of the acceptance region, i.e., in the critical region,

$\left(-\infty, \tilde{\mu} - \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}\right) \cup \left( \tilde{\mu} + \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}, \infty\right),$

• we reject the null hypothesis with $$\alpha \times 100\%$$ significance.
• Alternatively, if the sample mean lies within the acceptance region, we fail to reject the null hypothesis with $$\alpha\times 100\%$$ significance.

### General hypothesis testing of the mean

• We similarly will compute the P value in the case where $$\sigma^2$$ is unknown.

• Our test statistic will be the value,

\begin{align} t_0 = \frac{\overline{X} - \tilde{\mu}}{\frac{s}{\sqrt{n}}} \end{align}

• Using the t distribution in $$n-1$$ degrees of freedom as our probability model, we calculate the probability of observing a value at least as extreme as $$T_0$$.

• If the P value falls below $$\alpha$$, we can reject the null hypothesis at $$\alpha\times 100\%$$ significance.

• This is how the t.test() function computes both the confidence interval and the hypothesis test simultaneously.

• In particular, it will assume a null hypothesis of $$H_0: \mu = 0$$ be default.

• This default value is adjusted in the function keywords, along with the alternate hypothesis.
• We will now begin to consider how to use this function more generally.

### General hypothesis testing of the mean – examples

• The sodium content of twenty 300-gram boxes of organic cornflakes was determined.

• The data (in milligrams) are as follows:

sodium_sample <- c(131.15, 130.69, 130.91, 129.54, 129.64, 128.77,130.72, 128.33, 128.24, 129.65, 130.14, 129.29, 128.71, 129.00, 129.39, 130.42, 129.53, 130.12, 129.78, 130.92)

• Let's suppose we want to test the hypothesis,

\begin{align} H_0: \mu = 130 & & H_1:\mu \neq 130; \end{align}

• If we use t.test() directly, notice the output

t.test(sodium_sample)


One Sample t-test

data:  sodium_sample
t = 662.06, df = 19, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
129.3368 130.1572
sample estimates:
mean of x
129.747


### General hypothesis testing of the mean – examples

• Rather, to set the correct null and alternative hypothesis, we write,
t.test(sodium_sample, mu=130, alternative="two.sided")


One Sample t-test

data:  sodium_sample
t = -1.291, df = 19, p-value = 0.2122
alternative hypothesis: true mean is not equal to 130
95 percent confidence interval:
129.3368 130.1572
sample estimates:
mean of x
129.747

• Notice that the above includes the test statistic $$t_0 = -1.291$$.

• This also lists the number of degrees of freedom $$df= n-1 = 19$$ for the t distribtion.
• Most importantly, this lists the P value, $$\approx 0.2122$$.

• If we take $$\alpha=0.05$$, a common convention, we say $$P> \alpha$$, such that we fail to reject the null hypothesis of $$\mu = 130$$.

### General hypothesis testing of the mean – examples

• Suppose we wanted to perform a hypothesis test to make sure the mean sodium is not too high;

• if we wanted to evaluate the one-sided hypothesis test

\begin{align} H_0: \mu \leq 130 & & H_1:\mu >130, \end{align}

• we would write in R

t.test(sodium_sample, mu=130, alternative="greater")


One Sample t-test

data:  sodium_sample
t = -1.291, df = 19, p-value = 0.8939
alternative hypothesis: true mean is greater than 130
95 percent confidence interval:
129.4081      Inf
sample estimates:
mean of x
129.747

• Here, we once again fail to reject the null hypothesis at $$\alpha\times 100\% = 5\%$$ significance, as $$P\approx 0.89$$.

### General hypothesis testing of the mean – examples

• Computing the power of a t test is about the same as if we were computing the power with a normal distribution, replacing the appropriate z / t critical values and probabilities as necessary.

• However, computing sample size necessary for a hypothesis test to reach a certain power is very complicated, and is much more easily done with technology.

• Moreover, there is a built-in feature in R that will compute either the power of a test, or the needed sample size to attain a power, with the t test.

• The power.t.test() takes the following arguments

power.t.test(n, delta, sd, sig.level, power, alternative, type="one.sample")

• where
• n is the sample size
• delta is the difference between the assumed, but untrue, null hypothesis and the unknown, but assumed true, alternative hypothesis;
• sd is the the sample standard deviation;
• sig.level is the value of $$\alpha$$;
• power is the power of the test;
• alternative is the alternative hypothesis; and
• we need to specify the type="one.sample" as above.

### General hypothesis testing of the mean – examples

• When we enter the power.t.test(),
power.t.test(n, delta, sd, sig.level, power, alternative, type="one.sample")

• we will actually leave out one of power or n out as an argument.

• The argument that is left out, power or n, will be computed from the other arguments.

• We will continue our example with the sodium sample, now evaluating the power of our earlier tests

t.test(sodium_sample, mu=130, alternative="two.sided")


One Sample t-test

data:  sodium_sample
t = -1.291, df = 19, p-value = 0.2122
alternative hypothesis: true mean is not equal to 130
95 percent confidence interval:
129.3368 130.1572
sample estimates:
mean of x
129.747


### General hypothesis testing of the mean – examples

• In our sodium sample example, we had
s <- sd(sodium_sample)
n <- length(sodium_sample)
mu_null <- 130.0

• Suppose we have a specific value for the alternative hypothesis in mind, i.e.,
mu_alternative <- 130.5

• and we wish to determine the power of the test to reject the false, null hypothesis.

• We will leave the power argument blank in the function, but we need to calculate delta.

• delta is given as the absolute difference between our false null hypothesis, and the true alternative, i.e.,

delta <- abs(mu_null - mu_alternative)
delta

[1] 0.5


### General hypothesis testing of the mean – examples

• To calculate the power of the hypothesis test,

\begin{align} H_0 : \mu = 130 & & H_1:\mu \neq 130 \end{align}

• where we assume the true alternative hypothesis is $$H_1: \mu=130.5$$,

• with a significance level of $$\alpha=0.05$$,

• we can compute this at once witht the power.t.test() as:

power.t.test(n=n, delta=delta, sd=s, sig.level=0.05, power=NULL, type="one.sample")


One-sample t test power calculation

n = 20
delta = 0.5
sd = 0.8764288
sig.level = 0.05
power = 0.6775708
alternative = two.sided


### General hypothesis testing of the mean – examples

• Suppose we want to calculate power of the same type of hypothesis test, but with a different, one-sided alternative hypothesis.

• e.g.,

\begin{align} H_0:\mu \leq 130 & & H_1 :\mu > 130. \end{align}

• We specify this in the function as,

power.t.test(n=n, delta=delta, sd=s, alternative="one.sided" , sig.level=0.05, power=NULL, type="one.sample")


One-sample t test power calculation

n = 20
delta = 0.5
sd = 0.8764288
sig.level = 0.05
power = 0.7921742
alternative = one.sided


### General hypothesis testing of the mean – examples

• On the other hand, suppose we need to find the sample size necessary to meet a certain power with one of the earlier hypothesis tests.

• E.g., we might try to reject the null if a true mean sodium content is actually 130.1 milligrams, with a power of the test equal to 0.75.

• To do so, we now need to negelct the sample size argument n and supply the power argument power.

• The needed arguments are assigned below:

s <- sd(sodium_sample)
mu_null <- 130.0
mu_alternative <- 130.1
delta <- abs(mu_null - mu_alternative)
pow <- 0.75


### General hypothesis testing of the mean – examples

• We determine the appropriate sample size via
power.t.test(n=NULL, delta=delta, sd=s, power=pow, type="one.sample")


One-sample t test power calculation

n = 535.0307
delta = 0.1
sd = 0.8764288
sig.level = 0.05
power = 0.75
alternative = two.sided

• for the two-sided test, or for the one-sided test we use
power.t.test(n=NULL, delta=delta, sd=s, alternative="one.sided", power=pow, type="one.sample")


One-sample t test power calculation

n = 414.5589
delta = 0.1
sd = 0.8764288
sig.level = 0.05
power = 0.75
alternative = one.sided


## Summary

• This covers all the standard tools for general hypothesis testing and confidence intervals for the mean;

• the last homework exercises and the final project questions can be solved with these tools.
• Other hypothesis tests, such as a test on the standard deviation,

\begin{align} H_0 :\sigma = \tilde{\sigma} & & H_1: \sigma \neq \tilde{\sigma} \end{align} can be performed very similarly in R, using built-in functions.

• For further hypothesis testing in the future, you are recommended to search the appropriate R documentation for how to use these functions.

• The interpretations of the P value, level of significance, power of the test will all translate;

• however, the specific distribution / test statistic will change in general.
• However, the test statistic, P value and power are all computed “under-the-hood” and it is more important that you can comfortably interpret these outputs.

• The objective for this course was to give all students a working toolbox for standard statistical analysis and inference.

• Specifically, we learned the underlying theory of statistical inference, and how to make such an analysis in the modern environment of R Markdown.

• Everyone is encouraged to continue building their skills in this framework;

• for a more detailed class on statistical computation, consider STAT 445 in the fall.