When calculating probabilities using binomial expansions, we can calculate these probabilities for an individual value (\(P(x = a)\)) or a cumulative value \(P(x<a), \space P(x\leq a), \space P(x\geq a)\).
In hypothesis testing, we are testing as to whether or not these calculated probabilities can lead us to accept or reject a hypothesis.
We will be focusing on regions of binomial distribution; therefore, we are looking at cumulative values.
Types of hypotheses
There are two main types of hypotheses:
The null hypothesis (H0) is the hypothesis we assume happens, and it assumes there is no difference between certain characteristics of a population. Any difference is purely down to chance.
The alternative hypothesis (H1) is the hypothesis we can try to prove using the data we have been given.
We can either:
What are the steps to undertake a hypothesis test?
There are some key terms we need to understand before we look at the steps of hypothesis testing:
Critical value – this is the value where we go from accepting to rejecting the null hypothesis.
Critical region – the region where we are rejecting the null hypothesis.
Significance Level – a significance level is the level of accuracy we are measuring, and it is given as a percentage. When we find the probability of the critical value, it should be as close to the significance level as possible.
One-tailed test – the probability of the alternative hypothesis is either greater than or less than the probability of the null hypothesis.
Two-tailed test – the probability of the alternative hypothesis is just not equal to the probability of the null hypothesis.
So when we undertake a hypothesis test, generally speaking, these are the steps we use:
STEP 1 – Establish a null and alternative hypothesis, with relevant probabilities which will be stated in the question.
STEP 2 – Assign probabilities to our null and alternative hypotheses.
STEP 3 – Write out our binomial distribution.
STEP 4 – Calculate probabilities using binomial distribution. (Hint: To calculate our probabilities, we do not need to use our long-winded formula, but in the Casio Classwiz calculator, we can go to Menu -> Distribution -> Binomial CD and enter n as our number in the sample, p as our probability, and X as what we are trying to calculate).
STEP 5 – Check against significance level (whether this is greater than or less than the significance level).
STEP 6 – Accept or reject the null hypothesis.
Let's look at a few examples to explain what we are doing.
One-tailed test example
As stated above a one-tailed hypothesis test is one where the probability of the alternative hypothesis is either greater than or less than the null hypothesis.
A researcher is investigating whether people can identify the difference between Diet Coke and full-fat coke. He suspects that people are guessing. 20 people are selected at random, and 14 make a correct identification. He carries out a hypothesis test.
a) Briefly explain why the null hypothesis should be H0, with the probability p = 0.5 suggesting they have made the correct identification.
b) Complete the test at the 5% significance level.
SOLUTION:| Step | Example |
| STEP 1 - Establish a null and an alternative hypothesis, with relevant probabilities, which will be stated in the question. | Null hypothesis (H0): People are guessing, so they have an equal chance of making a correct or incorrect identification. Alternative hypothesis H1: People are not guessing and know how to tell the difference. |
STEP 2 - Assign probabilities to our null and alternative hypotheses. | \(\begin{align} H_0: p = 0.5 \\ H_1: p > 0.5\end{align}\)The reason for our probability for H1 is because to disprove people are guessing, more people need to get it right than wrong. |
| STEP 3 - Write out our binomial distribution. | There are 20 people, and the probability of the null hypothesis is 0.5 therefore if we call our event X:\(X \sim B(20,0.5)\) |
| STEP 4 - Calculate probabilities using binomial distribution. | So 14 people made the correct identification; therefore, we need to calculate \(P(X\geq 14) = 0.05765914916\)The reason we use the greater than or equal to sign is because \(p > 0.5\). |
| STEP 5 - Check against significance level (greater than or less than the significance level). | \(0.05765914916 > 0.05\), so it is not in the critical region as it would be less than 0.05 if that were the case. |
| STEP 6 - Accept or reject the null hypothesis. | As it is not in the critical region, we make the conclusion that we accept the null hypothesis. |
Two-tailed test example
In a two-tailed test, the probability of our alternative hypothesis is just not equal to the probability of the null hypothesis.
A coffee shop provides free espresso refills. The probability that a randomly chosen customer uses these refills is stated to be 0.35. A random sample of 20 customers is chosen, and 9 of them have used the free refills.
Carry out a hypothesis test to a 5% significance level to see if the probability that a randomly chosen customer uses the refills is different to 0.35.
SOLUTION: | Step | Example |
| STEP 1 - Establish a null and an alternative hypothesis, with relevant probabilities, stated in the question. | Null Hypothesis H0: Only that percentage of people will use the free espresso refills. Alternative hypothesis H1: Either more or fewer people will use the free espresso refills. |
| STEP 2 - Assign probabilities to our null and alternative hypotheses. | \(\begin{align} H_0: p = 0.35 \\ H_1: p \ne 0.35 \end{align}\)As this is a two-tailed test, the probability of the alternative hypothesis is just different to 0.35. |
| STEP 3 - Write out our binomial distribution. | There are 20 people, and the probability of the null hypothesis is 0.35 therefore if we call our event X: \(X \sim B(20,0.35)\) |
| STEP 4 - Calculate probabilities using binomial distribution. | This time as \(p \ne 0.35\), we need to calculate both \(P(X \leq 9)\) and \(P(X \geq 9)\), as in a two-tailed test, there are two critical regions.\(\begin{align} P(X \leq 9) = 0.8782194139 \\ P(X \geq 9) = 0.2376223533 \end{align}\) |
| STEP 5 - Check against significance level (greater than or less than the significance level). | As this is a two-tailed test, we need to divide the significance level into 2, so we compare against 0.025 instead of 0.05 and \(\begin{align}0.8782194139 > 0.025\\ 0.2376223533 > 0.025 \end{align}\) |
| STEP 6 - Accept or reject the null hypothesis. | Neither of these falls into our critical region (although one is far closer than the other) meaning we accept our null hypothesis. |
So our key difference with two-tailed tests is that we compare the value to half the significance level rather than the actual significance level.
Critical values and critical regions
Remember from earlier critical values are the values in which we move from accepting to rejecting the null hypothesis. A binomial distribution is a discrete distribution; therefore, our value has to be an integer.
You have a large number of statistical tables in the formula booklet that can help us find these; however, these are inaccurate as they give us exact values not values for the discrete distribution.
Therefore the best way to find critical values and critical regions is to use a calculator with trial and error till we find an acceptable value:
STEP 1 - Plug in some random values until we get to a point where for two consecutive values, one probability is above the significance level, and one probability is below.
STEP 2 - The one with the probability below the significance level is the critical value.
STEP 3 - The critical region, is the region greater than or less than the critical value.
Let's look at this through a few examples.
Worked examples for critical values and critical regions
A mechanic is checking to see how many faulty bolts he has. He is told that 30% of the bolts are faulty. He has a sample of 25 bolts. He believes that less than 30% are faulty. Calculate the critical value and the critical region.
SOLUTION:
Let's use the above steps to help us out.
| Step | Example |
| STEP 1 - Plug in some random values until we get to a point where for two consecutive values, one probability is above the significance level, and one probability is below. | So establishing our null and alternative hypotheses probabilities: \(\begin{align}H_0: p = 0.3 \\ H_1: p < 0.3 \end{align}\)Meaning we are calculating less than or equal to probabilities. If we try a few values: \(\begin{align}P(X \leq 5) = 0.1934884421 \\ P(X \leq 4) = 0.09047191855 \\ P(X \leq 3) = 0.03324051659 \end{align}\)We can see that \(P(X \leq 4)\) is above our 0.05 significance level and \(P(X \leq 3)\) is less than our 0.05 significance level. |
| STEP 2 - The one with the probability below the significance level is the critical value. | The lower value is our critical value so X = 3 is our critical value. |
| STEP 3 - The critical region is greater than or less than the critical value. | The critical region is the region less than the critical value so \(X \leq 3\). |
A teacher believes that 40% of the students watch TV for two hours a day. A student disagrees and believes that students watch either more or less than two hours. In a sample of 30 students, calculate the critical regions.
SOLUTION:
As this is a two-tailed test, there are two critical regions, one on the lower end and one on the higher end. Also, remember the probability we are comparing with is that of half the significance level.
| Step | Example |
| STEP 1 - Plug in some random values until we get to a point where for two consecutive values, one probability is above the significance level, and one probability is below. | So let's firstly establish our null and alternative hypotheses probabilities: \(\begin{align} H_0: p = 0.4 \\ H_1: p \ne 0.4 \end{align}\)Let's start looking at the lower end where\(\begin{align}&P(X \leq a): \\ &P(X \leq 5) = 0.005658796379 \\ &P(X \leq 6) = 0.01718302499 \\ &P(X \leq 7) = 0.0435241189 \end{align}\). And if we look at the upper end where \(P(X \geq a)\): \(\begin{align} P(X \geq 16) = 0.09705684391 \\ P(X \geq 17) = 0.04811171242 \\ P(X \geq 18) = 0.02123987608 \end{align}\) |
| STEP 2 - The one with the probability below the significance level is the critical value. | Remember, we're comparing with our significance level of 0.025, not 0.05. On the lower end: \(\begin{align}0.005658796379 > 0.025 \\0.01718302499 < 0.025 \end{align}\)So our critical value is X = 6.Similarly, on the upper end: \(\begin{align}0.04811171242 > 0.025 \\ 0.02123987608 < 0.025\end{align}\)So our critical value is X = 18. And therefore, our critical values are X = 6, X = 18. |
| STEP 3 - The critical region is the region greater than or less than the critical value. | Our critical regions are therefore: \(X \leq 6\) and \(X \geq 18\). |
Binomial Hypothesis Test - Key takeaways
- Hypothesis testing is the process of using binomial distribution to help us reject or accept null hypotheses.
- A null hypothesis is what we assume to be happening.
- If data disprove a null hypothesis, we must accept an alternative hypothesis.
- We use binomial CD on the calculator to help us shortcut calculating the probability values.
- The critical value is the value where we start rejecting the null hypothesis.
- The critical region is the region either below or above the critical value.
- Two-tailed tests contain two critical regions and critical values.
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