# Confidence intervals and an introduction to hypothesis testing

04/14/2021

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## Outline

• The following topics will be covered in this lecture:

• Confidence intervals for the mean, with unknown population standard deviation
• Large sample size
• Student t distribution
• Student t confidence intervals
• Student t confidence intervals with real-life methods
• An introduction to hypothesis testing
• Statistical hypotheses
• The null and alternative hypothesis
• Two-sided versus one-sided hypothesis tests

## Confidence intervals, variance unknown

• In practice, we almost never know the true population standard deviation $$\sigma$$ and we must use the sample standard deviation $$s$$ as a point estimate.

• Our standard error estimate is $$\hat{\sigma}_\overline{X}= \frac{s}{\sqrt{n}}$$, and this will be utilized for a more general construction of confidence intervals.

• If we have a large sample size, with $$n>40$$, we can use this estimate of the standard error effectively within the confidence interval as follows.

Large-Sample Confidence Interval on the Mean
When n is large, the quantity $\frac{\overline{X} - \mu}{\frac{s}{\sqrt{n}}}$ has an approximate standard normal distribution. Consequently, $x − z_\frac{\alpha}{2} \frac{s}{\sqrt{n}} \leq \mu \leq x + z_\frac{\alpha}{2} \frac{s}{\sqrt{n}}$ is a large-sample confidence interval for $$\mu$$, with confidence level of approximately $$(1-\alpha)\times 100\%$$.
• This is a form of the central limit theorem being used again where the underlying population distribution does not matter;

• the sampling distribution of the sample mean can be approximated with a normal assumption with a standard error $$\sigma_{\overline{X}}$$.
• If we estimate $$\sigma$$ with $$s$$, we can get an approximation of the normal using $$\hat{\sigma}_{\overline{X}}$$ as an approximation of the standard error.

### Confidence intervals, variance unknown – continued

• However, when the sample is small and $$\sigma^2$$ is unknown, we must make an assumption about the form of the underlying distribution to obtain a valid CI procedure.

• A reasonable assumption in many cases is that the underlying distribution is normal.

• Many populations encountered in practice are well approximated by the normal distribution, so this assumption will lead to confidence interval procedures of wide applicability.

• In fact, moderate departure from normality will have little effect on validity.

• When the assumption is unreasonable, an alternative is to use nonparametric statistical procedures that are valid for any underlying distribution.

### Confidence intervals, variance unknown – continued

• Suppose that the population of interest has a normal distribution with unknown mean $$\mu$$ and unknown variance $$\sigma^2$$.

• Assume that a random sample of size $$n$$, say, $$X_1, X_2 , \cdots , X_n$$, is available, and let $$\overline{X}$$ and $$S^2$$ be the sample mean and variance, respectively.

• We wish to construct a two-sided CI on $$\mu$$;

• if the variance $$\sigma$$ is known, we know that

$Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$

has a standard normal distribution.

• When $$\sigma$$ is unknown, we use the estimate $$\hat{\sigma}_\overline{X} = \frac{S}{\sqrt{n}}$$ for the standard error.
• The random variable $$Z$$ now becomes

$T = \frac{\overline{X} − \mu}{\frac{S}{\sqrt{n}}}.$

### Confidence intervals, variance unknown – continued

• For the random variable

$T = \frac{\overline{X} − \mu}{\frac{S}{\sqrt{n}}}.$

logical questions are:

• what is the distribution of $$T$$?; and
• is the distribution very different than the standard normal?
• If $$n$$ is large, the distribution differs very little from the standard normal by the central limit theorem.

• However, $$n$$ is usually small in most engineering problems, and in this situation, a different distribution must be employed to construct the CI.

• Let's suppose that we have a random sample $$X_1, \cdots, X_n$$ from a normal distribution with population mean $$\mu$$ and population standard deviation $$\sigma$$.
• The sample mean $$\overline{X}$$ and the sample standard deviation $$S$$ are computed from the above observations.
• Then, it is an extremely important and non-trivial result that the random variable, $\frac{\overline{x} - \mu}{\frac{S}{\sqrt{n}}}$ is distributed according to a student t with $$n-1$$ degrees of freedom.

### Student's t-distribution

• The pdf of the t-distribution is

\begin{align} f(T,n) = \frac{\Gamma\left\{\frac{n+1}{2}\right\}}{\sqrt{\pi n} \Gamma\left(\frac{n}{2}\right)\left(1 + \frac{T^2}{n}\right)^{\frac{n+1}{2}}} \end{align} where the Gamma function is a “special function”.

• We plot the density below fo n=1, n=2 and n=5 degrees of freedom with the normal density plotted for reference.

par(cex = 2.0, mar = c(5, 4, 4, 2) + 0.3)
t = seq(-5, 5, length = 300)
colors = c("black", "red", "green")
df = c(1, 2, 5)  # degrees of freedom(df) for the t-distribution
plot(t, dnorm(t, 0, 1), xlab = "t", ylab = "pdf", type = "l", lwd = 2, col="blue")
for (i in 2:4) {  lines(t, dt(t, df[i]), col = colors[i])}


### Student's t-distribution

• The degrees of freedom determine the shape of the student t.

• For $$n > 2$$ degrees of freedom, the mean and variance of Student’s t-distribution are

\begin{align} \mu_T= 0 & & \sigma_T^2 = \frac{n}{n-2} \end{align}

• As $$n\rightarrow \infty$$, the student t distribution becomes closer and eventually converges to the standard normal.

par(cex = 2.0, mar = c(5, 4, 4, 2) + 0.3)
t = seq(-5, 5, length = 300)
colors = c("black", "red", "green")
df = c(10, 100, 1000)  # degrees of freedom(df) for the t-distribution
plot(t, dnorm(t, 0, 1), xlab = "t", ylab = "pdf", type = "l", lwd = 2, col="blue")
for (i in 2:4) {  lines(t, dt(t, df[i]), col = colors[i])}


### Student's t-distribution

• The quantiles of a t-distributed rv $$T$$ are denoted by $$t_p$$, and, due to symmetry, $$t_p =−t_{1− p}$$.

• In R the generic functions for the t distribution are the following:

• dt(x, df) is the probability density function of the t distribution with df degrees of freedom.
• pt(q, df) is the cumulative density function of the t distribution with df degrees of freedom.
• rt(n, df) randomly generates a sample of size n from the t distribution with df degrees of freedom.
• qt(p, df) is the quantile function of the t distribution with df degrees of freedom.
• With these above generic functions for the t distribution, we can almost identically compute the student t confidence interval as we did for the normal confidence interval.

## Student t confidence intervals

• Formally, we will write
Confidence Interval on the Mean, Variance Unknown
If $$\overline{x}$$ and $$s$$ are the mean and standard deviation of a random sample from a normal distribution with unknown variance $$\sigma^2$$ with a sample size $$n$$. A $$(1-\alpha)\times 100\%$$ confidence interval on $$\mu$$ is given by \begin{align} &\overline{x} - \hat{\sigma}_\overline{X} t_\frac{\alpha}{2} \leq \mu \leq \overline{x} + \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}\\ \Leftrightarrow&\overline{x} - \frac{s}{\sqrt{n}} t_\frac{\alpha}{2} \leq \mu \leq \overline{x} + \frac{s}{\sqrt{n}}t_\frac{\alpha}{2} \end{align} where $$t_\frac{\alpha}{2}$$ is the upper $$\frac{\alpha}{2}$$ critical point of the t distribution with n − 1 degrees of freedom.
• In practice, this is how we will more typically compute a confidence interval on the mean.

• Because this is the most common way to compute such a confidence interval in practice, there are actually built-in functions in the R language to handle this.

• Computing the confidence interval “manually” with the quantile function is mostly pedagogical, but we will demonstrate how this is done with a few more examples.

• Shortly, we will learn how to compute confidence intervals and hypothesis tests with the t distribution using the

t.test()


function in R.

### Student t confidence intervals – continued

• For now, in order to compute the t confidence interval manually, we need to find the appropriate critical value for the equation

$\overline{x} - \frac{s}{\sqrt{n}} t_\frac{\alpha}{2} \leq \mu \leq \overline{x} + \frac{s}{\sqrt{n}}t_\frac{\alpha}{2}$

• We can find this critical point in the same way as for the normal, using R, as follows.

• Suppose we have a sample size of $$n=20$$; this gives $$n-1=19$$ degrees of freedom, i.e.,

t_alpha_over_2 <- qt(0.975, df=19)
t_alpha_over_2

[1] 2.093024


is the critical point for the $$95\%$$ two-sided confidence interval.

• The critical point for the $$99\%$$ two-sided confidence interval is given as
t_alpha_over_2 <- qt(0.995, df=19)
t_alpha_over_2

[1] 2.860935

• We will consider a full example of constructing the confidence interval in the following.

### Student t confidence intervals – example

• An article in the Journal of Materials Engineering describes the results of tensile adhesion tests;

• this is performed on the following U-700 alloy specimens, with the load at failure as follows (in megapascals):
alloy_load_failures <- c(19.8, 10.1, 14.9, 7.5, 15.4, 15.4, 15.4, 18.5, 7.9, 12.7, 11.9, 11.4, 11.4, 14.1, 17.6, 16.7, 15.8, 19.5, 8.8, 13.6, 11.9, 11.4)

• We can determine some key values as follows:
n <- length(alloy_load_failures)
n

[1] 22

x_bar <- mean(alloy_load_failures)
x_bar

[1] 13.71364

s <- sd(alloy_load_failures)
s

[1] 3.553576


### Student t confidence intervals – example

• Using the last values, we can compute the estimated standard error as
se <- s / sqrt(n)
se

[1] 0.7576249

• Notice, if we want to compute the $$95\%$$ confidence interval of the mean, we cannot use the z critical value accurately as the sample size is under 40, and we do not know the population standard deviation.

• Therefore, we compute the t critical value as

t_alpha_over_2 = qt(0.975, df=n-1)
t_alpha_over_2

[1] 2.079614

• Our corresponding t confidence interval is given as
ci <- c(x_bar - se * t_alpha_over_2, x_bar + se * t_alpha_over_2)
ci

[1] 12.13807 15.28920


### Student t confidence intervals – example

• Notice that
z_alpha_over_2 <- qnorm(0.975)
z_alpha_over_2

[1] 1.959964

• is smaller than
t_alpha_over_2

[1] 2.079614

• This demonstrates the way in which the t distribution models the increased uncertainty of the population mean, due to the unknown population standard deviation.

• As mentioned before, this process of manually computing confidence intervals is really just pedagogical.

• We will now begin to introduce the realistic way confidence intervals are computed in practice.

### Student t confidence intervals – example

• Recall, we have the following sample
alloy_load_failures

 [1] 19.8 10.1 14.9  7.5 15.4 15.4 15.4 18.5  7.9 12.7 11.9 11.4 11.4 14.1 17.6
[16] 16.7 15.8 19.5  8.8 13.6 11.9 11.4

• We can compute the confidence interval directly as
t.test(alloy_load_failures)


One Sample t-test

t = 18.101, df = 21, p-value = 2.731e-14
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
12.13807 15.28920
sample estimates:
mean of x
13.71364

• There are a few pieces of information above that we have yet to discuss – this is the t hypothesis test covered next.

### Student t confidence intervals – example

• Note, if we want to make a confidence interval for a different confidence level, we can supply an optional key word
t.test(alloy_load_failures, conf.level=0.99)


One Sample t-test

t = 18.101, df = 21, p-value = 2.731e-14
alternative hypothesis: true mean is not equal to 0
99 percent confidence interval:
11.56853 15.85874
sample estimates:
mean of x
13.71364

• In reality, this is the default way that one will compute a confidence interval on the mean.

• We will begin to favor this approach over the pedagogical approach, constructing confidence intervals using qt or qnorm.

• However, as the pedagogical approach emphasizes the mathematical concepts, we will still have a few exercises like this.
• The final project, in particular, will use both approaches.

## Hypothesis testing – motivation

• So far, we showed how a parameter of a population can be estimated from sample data;

• We first showed how to construct a point estimate based on a sample;

• however, a point estimate is is statistically unsatisfying due to the intrinsic uncertainty of this estimate due to sampling error.
• In order to rectify the issue with only providing a single point estimate, we constructed an interval of likely values called a confidence interval.

• With a level of confidence $$(1 -\alpha)\times 100\%$$, specified in terms of the failure rate $$\alpha$$, we supplied a range of plausible values for the parameter given the sample on hand.

• In many situations, a dual type of problem is of interest, where we will be concerned with how unlikely a possible parameter value might be.

• For a $$95\%$$ level of confidence, we had an $$\alpha=5\%$$ rate of failure in the confidence interval proceedure.

• This principle has been the basis of us finding $$z_\frac{\alpha}{2}$$ and $$t_\frac{\alpha}{2}$$ critical values for $$\alpha = 0.05$$ corresponding to $$5\%$$.

• Particularly, we would have found it unlikely that in more than $$1$$ out of $$20$$ replications of samples the associated confidence interval did not contain the true parameter.

## Hypothesis testing – motivation

• We can think of rephrasing the above principle as well:
• Suppose we are estimating the population mean $$\mu$$, and we have some hypothesis as to what the value might be, $$\tilde{\mu}$$.
• Let us suppose we created a $$95\%$$ confidence interval, $\left(\overline{X} - \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}, \overline{X} + \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}\right)$ and upon comparing with some realization $$\overline{x}$$ we found that $$\tilde{\mu}$$ was not in this region.
• If we are following the procedure correctly, and if the $$\tilde{\mu}$$ was actually equal to the true population $$\mu$$, then $\tilde{\mu} \text{ not in } \left(\overline{X} - \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}, \overline{X} + \hat{\sigma}_\overline{X} t_\frac{\alpha}{2}\right)$ only $$5\%$$ of the time.
• If we were to find that $$\tilde{\mu}$$ was actually in our confidence intervals in far less than $$5\%$$ of replications, this should question if $$\tilde{\mu}$$ was really a good hypothesis for the true $$\mu$$.
• In this sense, $$\alpha$$ represents a kind of criterion if we should question if a proposed value of $$\tilde{\mu}$$ is really appropriate.

### Hypothesis testing – motivation

• In real applications, there may be two competing claims (or hypotheses) about the value of a parameter.

• The engineer must use sample data to determine which claim is most plausible, and which one can be rejected as unlikely.

• For example, suppose that an engineer is designing an air crew escape system;

• this will consist of an ejection seat and a rocket motor that powers the seat.
• The rocket motor contains a propellant, and for the ejection seat to function properly, the propellant should have a mean burning rate of 50 cm/sec.

• If the burning rate is too low, the ejection seat may not function properly, leading to an unsafe ejection and possible injury of the pilot.

• Higher burning rates may imply instability in the propellant or an ejection seat that is too powerful, again leading to possible pilot injury.

• The practical engineering question that must be answered is: Does the mean burning rate of the propellant equal 50 cm/sec, or is it some other value (either higher or lower)?

• This type of question can be answered using a statistical technique called hypothesis testing.

• We have already gotten some idea of the duality of these problems as t.test() computes both simultaneously.

• We will now develop this idea more formally.

### Hypothesis testing – introduction

• Formally, we will define
Statistical Hypothesis
A statistical hypothesis is a statement about the parameters of one or more populations.
• Because we use probability distributions to model populations, a statistical hypothesis may also be thought of as a statement about the probability distribution of a random variable.

• The hypothesis will usually involve one or more parameters of this distribution.

• For example, consider the air crew escape system described already.

• Suppose that we are interested in the burning rate of the solid propellant.

• Burning rate is a random variable that can be described by a probability distribution.

• Suppose that our interest focuses on the mean burning rate (a parameter of this distribution).

• Specifically, we are interested in deciding whether or not the mean burning rate is $$50$$ centimeters per second.

• We may express this formally as

\begin{align} H_0∶& \mu = 50 \text{ centimeters per second}\\ H_1∶& \mu \neq 50 \text{ centimeters per second} \end{align}

• $$H_0$$ is known as the null hypothesis and $$H_1$$ is known as the alternative hypothesis.

### Hypothesis testing – introduction

• In hypothesis testing, the null and alternative hypotheses have special meanings philosophically and in the mathematics.

• We cannot generally “prove” a hypothesis to be true;

• generically, we will assume that the true population parameter is unobservable.
• Instead, we can only determine if a hypothesis seems unlikely enough to reject;

• this is similar to finding that our proposed parameter value was in far-fewer confidence intervals than predicted by the procedure.
• To begin such a test formally, we need to first make some assumption about the true parameter.

• This always takes the form of assuming the null hypothesis $$H_0$$.
• The null hypothesis $$H_0$$ will always take the form of an equality, or an inclusive inequality.

• That is, we take

\begin{align} H_0: & \theta \text{ is } (= / \leq / \geq) \text{ some proposed value}. \end{align}

• In our example, we wrote

\begin{align} H_0∶ & \mu = 50 \text{ centimeters per second}. \end{align}

### Hypothesis testing – introduction

• The contradictory / competing hypothesis is the alternative hypothesis, written

\begin{align} H_1: & \theta \text{ is } (\neq / > / <) \text{ some proposed value} \end{align}

• In our example, we wrote

\begin{align} H_1∶ & \mu \neq 50 \text{ centimeters per second}. \end{align}

• Once we have formed a null and alternative hypothesis:

\begin{align} H_0: & \theta \text{ is } (= / \leq / \geq) \text{ some proposed value}\\ H_1: & \theta \text{ is } (\neq / > / <) \text{ some proposed value} \end{align}

• we use the sample data to consider how likely or unlikely it was to observe such data with the proposed parameter.

• If the sample doesn't seem to fit the proposed parameter value, we deem the null hypothesis unlikely.
• If the null hypothesis is sufficiently unlikely, we reject the null hypothesis in favor of the alternative hypothesis.

• However, if the evidence (the sample) doesn't contradict the null hypothesis, we tentatively keep this assumption.

• This has not proven this assumption, it has only said that the hypothesis is not unlikely given our evidence.
• In our example, we would say either:

1. we reject the null hypothesis of $$H_0∶ \mu = 50$$ in favor of the alternative $$H_1: \mu \neq 50$$; or
2. we fail to reject the null hypothesis of $$H_0:\mu = 50$$.

### Hypothesis testing – introduction

• In our example, the alternative hypothesis specifies values of $$\mu$$ that could be either greater or less than 50 centimeters per second;

• therefore, it is called a two-sided alternative hypothesis.
• In some situations, we may wish to formulate a one-sided alternative hypothesis, as in

\begin{align} H_0∶ & \mu \geq 50\text{ centimeters per second} \\ H_1∶ & \mu < 50\text{ centimeters per second} \end{align}

• or

\begin{align} H_0∶ & \mu \leq 50\text{ centimeters per second} \\ H_1∶ & \mu > 50\text{ centimeters per second} \end{align}

• The above situations have an exact analogy with one-sided confidence bounds, similar to the two-sided test and the two-sided confidence interval.

• We will now elaborate on the meaning of determining if a hypothesis is sufficiently unlikely.

• This is directly related to the value $$\alpha$$ we used as a rate of failure for confidence intervals.