What two assumptions are made when conducting a (pooled two-sample) t -test?

When conducting a (pooled two-sample) t -test you need to assume that: the underlying distribution for each variable must be normal , the variances for the two groups are equal .

When would you use a one-tailed test to compare the difference between the means of two normal distributions?

You would use a one-tailed test to compare the difference between the means of two normal distributions when you want to test one of two following hypotheses: The population mean of one normal distribution is greater than the population mean of another normal distribution. The population mean of one normal distribution is smaller than the population mean of another normal distribution.

When would you use a two-tailed test to compare the difference between the means of two normal distributions?

You would use a two-tailed test to compare the difference between the means of two normal distributions when you want to test whether the population means of two normal distributions are not equal .

In an exam, how do you calculate the p -value for a two-sample t -test ?

In an exam, to calculate the p -value for a two-sample t -test you: input the data from the sample of the first population in one list on your GDC, input the data from the sample of the second population in another list on your GDC, select the pooled two-sample t -test option, choose the form of the alternative hypothesis, run the test.

True or False? If the p- value of a two-tailed t -test is greater than the significance level , then the means of two normal populations are equal .

False. If the p- value of a two-tailed t -test is greater than the significance level then the test concludes that there is insufficient evidence to say that the means are different . This does not guarantee that the means are equal .

True or False? A paired t -test can be thought of as a one-sample t -test .

True. A paired t -test can be thought of as a one-sample t -test .

When should you use a paired t -test ?

You should use a paired t -test when you have two sets of data from the same sample and you are testing the difference in the values. For example, you might be comparing the test scores of a class before and after revision.

True or False? A t -test can be used to test whether there is a linear correlation between two normally distributed variables.

True. A t -test can be used to test whether there is a linear correlation between two normally distributed variables.

What is the difference between a one-tailed test and a two-tailed test for linear correlation ?

A one-tailed test for linear correlation tests specifically for a particular type of correlation (positive or negative). Whereas, a two-tailed test just tests for any linear correlation . A two-tailed test is used when you do not know what type of correlation there might be.

What would the conclusion be if the p -value for a two-tailed test for linear correlation is less than the significance level ?

If the p -value for a two-tailed test for linear correlation is less than the significance level , then there is sufficient evidence to suggest that there is a linear correlation between the two variables. Note that the test does not determine whether it is a positive or negative correlation.

What would the conclusion be if the p -value for a test for positive linear correlation is greater than the significance level ?

If the p -value for a test for positive linear correlation is greater than the significance level , then there is insufficient evidence to suggest that there is a positive linear correlation between the two variables.

What is a Type I error ?

A Type I error occurs if the null hypothesis is rejected when it is true .

What is a Type II error ?

A Type I error occurs if the null hypothesis is not rejected when it is not true .

True or False? The probability of a Type I error is always equal to the stated significance level .

False. The probability of a Type I error is always less than or equal to the stated significance level . For continuous distributions , the probability is equal to the stated significance level.

True or False? The critical region will maximise the probability of a Type I error while keeping it less than or equal to the stated significance level .

True. The critical region will maximise the probability of a Type I error while keeping it less than or equal to the stated significance level .

If the null hypothesis is not rejected , what type of error could have been made?

If the null hypothesis is not rejected , then a Type II error could have been made.

The probability of which type of error usually decreases when the significance level is increased ?

The probability of a Type II error usually decreases when the significance level is increased .

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Hypothesis Testing for Population Parameters (DP IB Applications & Interpretation (AI))

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If you are testing the mean of a normal population, when do you use the normal distribution (z-test)?

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If you are testing the mean of a normal population, when do you use the normal distribution (z-test)?
If you are testing the mean of a normal population, you use the normal distribution (z-test) if the population variance, $σ^{2}$ , is known.
If you are testing the mean of a normal population, when do you use the t-distribution (t-test)?
If you are testing the mean of a normal population, you use the t-distribution (t-test) if the population variance, $σ^{2}$ , is unknown.
Using symbols, what does the alternative hypothesis look like for a two-tailed test for the population mean of a normal distribution?
The alternative hypothesis for a two-tailed test for the population mean of a normal distribution looks like $H_{1} : μ \neq μ_{0}$ , where $μ_{0}$ is the assumed mean.
True or False?
If $H_{0} : μ = μ_{0}$ and $H_{1} : μ > μ_{0}$ are used to test the mean of a normal distribution with variance $σ^{2}$ , then the p-value for a sample mean $\bar{x}$ taken from a sample of $n$ observations is $P (\bar{X} > \bar{x})$ , where $\bar{X} ~ N (μ_{0}, \frac{σ^{2}}{n})$ .
True.
If $H_{0} : μ = μ_{0}$ and $H_{1} : μ > μ_{0}$ are used to test the mean of a normal distribution with variance $σ^{2}$ , then the p-value for a sample mean $\bar{x}$ taken from a sample of $n$ observations is $P (\bar{X} > \bar{x})$ , where $\bar{X} ~ N (μ_{0}, \frac{σ^{2}}{n})$ .
The hypotheses, $H_{0} : μ = 5$ and $H_{1} : μ \neq 5$ , are used to test the mean of a normal distribution with variance $3^{2}$ .
A sample of 100 observations is taken and the sample mean is $4$ .
How would you find the p-value?
The hypotheses, $H_{0} : μ = 5$ and $H_{1} : μ \neq 5$ , are used to test the mean of a normal distribution with variance $3^{2}$ .
A sample of 100 observations is taken and the sample mean is $4$ .
The p-value is equal to $2 \times P (\bar{X} < 4)$ , where $\bar{X} ~ N (5, \frac{3^{2}}{100})$ .
The probability is doubled as it is a two-tailed test.
The hypotheses, $H_{0} : μ = 5$ and $H_{1} : μ < 5$ , are used to test the mean of a normal distribution with variance $3^{2}$ .
A sample of 100 observations is taken and the sample mean is calculated.
How would you find the critical region if the significance level is 5%?
The hypotheses, $H_{0} : μ = 5$ and $H_{1} : μ < 5$ , are used to test the mean of a normal distribution with variance $3^{2}$ .
A sample of 100 observations is taken and the sample mean is calculated.
The critical region is $\bar{X} < c$ where $c$ is found using the inverse normal distribution with $P (\bar{X} < c) = 0.05$ , where $\bar{X} ~ N (5, \frac{3^{2}}{100})$ .
The hypotheses, $H_{0} : μ = μ_{0}$ and $H_{1} : μ \neq μ_{0}$ , are used to test the mean of a normal distribution with variance $σ^{2}$ .
A sample of $n$ observations is taken and the sample mean is calculated.
The critical regions for a 5% level of significance are $\bar{X} < 4$ and $\bar{X} > 6$ .
What is the value of $P (\bar{X} < 4)$ , where $\bar{X} ~ N (μ_{0}, \frac{σ^{2}}{n})$ ?
The hypotheses, $H_{0} : μ = μ_{0}$ and $H_{1} : μ \neq μ_{0}$ , are used to test the mean of a normal distribution with variance $σ^{2}$ .
A sample of $n$ observations is taken and the sample mean is calculated.
The critical regions for a 5% level of significance are $\bar{X} < 4$ and $\bar{X} > 6$ .
$P (\bar{X} < 4) = \frac{0.05}{2} = 0.025$ , where $\bar{X} ~ N (μ_{0}, \frac{σ^{2}}{n})$ .
$H_{0} : μ = μ_{0} H_{1} : μ \neq μ_{0}$
What would the conclusion be if the null hypothesis is not rejected?
$H_{0} : μ = μ_{0} H_{1} : μ \neq μ_{0}$
If the null hypothesis is not rejected then there is insufficient evidence to suggest that the mean is different to $μ_{0}$ .
$H_{0} : μ = μ_{0} H_{1} : μ \neq μ_{0}$
What would the conclusion be if the null hypothesis is rejected?
$H_{0} : μ = μ_{0} H_{1} : μ \neq μ_{0}$
If the null hypothesis is rejected then there is sufficient evidence to suggest that the mean is different to $μ_{0}$ .
If you are testing the difference between the means of two normal populations, when do you use the normal distribution (z-test)?
If you are testing the difference between the means of two normal populations, you use the normal distribution (z-test) if the population variance, $σ^{2}$ , is known.
If you are testing the difference between the means of two normal populations, when do you use the t-distribution (t-test)?
If you are testing the difference between the means of two normal populations, you use the t-distribution (t-test) if the population variance, $σ^{2}$ , is unknown.
What two assumptions are made when conducting a (pooled two-sample) t-test?
When conducting a (pooled two-sample) t-test you need to assume that:
- the underlying distribution for each variable must be normal,
- the variances for the two groups are equal.
When would you use a one-tailed test to compare the difference between the means of two normal distributions?
You would use a one-tailed test to compare the difference between the means of two normal distributions when you want to test one of two following hypotheses:
1. The population mean of one normal distribution is greater than the population mean of another normal distribution.
2. The population mean of one normal distribution is smaller than the population mean of another normal distribution.
When would you use a two-tailed test to compare the difference between the means of two normal distributions?
You would use a two-tailed test to compare the difference between the means of two normal distributions when you want to test whether the population means of two normal distributions are not equal.
In an exam, how do you calculate the p-value for a two-sample t-test?
In an exam, to calculate the p-value for a two-sample t-test you:
- input the data from the sample of the first population in one list on your GDC,
- input the data from the sample of the second population in another list on your GDC,
- select the pooled two-sample t-test option,
- choose the form of the alternative hypothesis,
- run the test.
Using symbols, what does the alternative hypothesis look like for a two-tailed test for the difference between the means of two normal distributions?
The alternative hypothesis for a two-tailed test for the difference between the means of two normal distributions looks like $H_{1} : μ_{X} \neq μ_{Y}$ .
$H_{0} : μ_{X} = μ_{Y} H_{1} : μ_{X} < μ_{Y}$
What would the conclusion be if the null hypothesis is not rejected?
$H_{0} : μ_{X} = μ_{Y} H_{1} : μ_{X} < μ_{Y}$
If the null hypothesis is not rejected then there is insufficient evidence to suggest that the mean of population Y is greater than the mean of population X.
$H_{0} : μ_{X} = μ_{Y} H_{1} : μ_{X} < μ_{Y}$
What would the conclusion be if the null hypothesis is rejected?
$H_{0} : μ_{X} = μ_{Y} H_{1} : μ_{X} < μ_{Y}$
If the null hypothesis is rejected then there is sufficient evidence to suggest that the mean of population Y is greater than the mean of population X.
True or False?
If the p-value of a two-tailed t-test is greater than the significance level, then the means of two normal populations are equal.
False.
If the p-value of a two-tailed t-test is greater than the significance level then the test concludes that there is insufficient evidence to say that the means are different.
This does not guarantee that the means are equal.
True or False?
A paired t-test can be thought of as a one-sample t-test.
True.
A paired t-test can be thought of as a one-sample t-test.
When should you use a paired t-test?
You should use a paired t-test when you have two sets of data from the same sample and you are testing the difference in the values.
For example, you might be comparing the test scores of a class before and after revision.
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p < p_{0}$ are used to test the proportion using a binomial distribution.
A random sample of $n$ observations is taken and the number of successes, $x$ , is recorded.
How do you calculate the p-value?
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p < p_{0}$ are used to test the proportion using a binomial distribution.
A random sample of $n$ observations is taken and the number of successes, $x$ , is recorded.
The p-value is $P (X \leq x)$ , where $X ~ B (n, p_{0})$ .
True or False?
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p > p_{0}$ are used to test the proportion using a binomial distribution.
A random sample of $n$ observations is taken and the number of successes, $x$ , is recorded.
The p-value is $P (X > x)$ , where $X ~ B (n, p_{0})$ .
False.
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p > p_{0}$ are used to test the proportion using a binomial distribution.
A random sample of $n$ observations is taken and the number of successes, $x$ , is recorded.
The p-value is $P (X \geq x)$ , where $X ~ B (n, p_{0})$ .
You need to remember to include the value $x$ in the inequality.
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p > p_{0}$ are used to test the proportion using a binomial distribution.
A random sample of $n$ observations is taken and the number of successes, $x$ , is recorded.
How do you find the critical region when a 5% level of significance is used?
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p > p_{0}$ are used to test the proportion using a binomial distribution.
To find the critical region when a 5% level of significance is used, you find the smallest value $c$ such that $P (X \geq c) \leq 0.05$ , where $X ~ B (n, p_{0})$ .
This means $P (X \geq c - 1) > 0.05$ , where $X ~ B (n, p_{0})$ .
True or False?
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p < p_{0}$ are used to test the proportion using a binomial distribution.
A random sample of $n$ observations is taken and the number of successes, $x$ , is recorded.
The critical value is the smallest value, $c$ , such that $P (X \leq c)$ , where $X ~ B (n, p_{0})$ , is less than the significance level.
False.
The hypotheses $H_{0} : p = p_{0}$ and $H_{1} : p < p_{0}$ are used to test the proportion using a binomial distribution.
A random sample of $n$ observations is taken and the number of successes, $x$ , is recorded.
The critical value is the largest value, $c$ , such that $P (X \leq c)$ , where $X ~ B (n, p_{0})$ , is less than the significance level.
The probability that a coin lands on tails is $p$ .
What would the hypotheses be if you wanted to test whether the coin is fair?
The probability that a coin lands on tails is $p$ .
If you wanted to test whether the coin is fair you could use the hypotheses $H_{0} : p = \frac{1}{2}$ and $H_{1} : p \neq \frac{1}{2}$ .
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m > m_{0}$ are used to test the mean using a Poisson distribution.
A random interval is observed and the number of occurrences, $x$ , is recorded.
How do you calculate the p-value?
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m > m_{0}$ are used to test the mean using a Poisson distribution.
A random interval is observed and the number of occurrences, $x$ , is recorded.
The p-value is $P (X \geq x)$ , where $X ~ Po (m_{0})$ .
True or False?
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m < m_{0}$ are used to test the mean using a Poisson distribution.
A random interval is observed and the number of occurrences, $x$ , is recorded.
The p-value is $P (X \leq x)$ , where $X ~ Po (m_{0})$ .
True.
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m < m_{0}$ are used to test the mean using a Poisson distribution.
A random interval is observed and the number of occurrences, $x$ , is recorded.
The p-value is $P (X \leq x)$ , where $X ~ Po (m_{0})$ .
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m < m_{0}$ are used to test the mean using a Poisson distribution.
A random interval is observed and the number of occurrences, $x$ , is recorded.
How do you find the critical region when a 1% level of significance is used?
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m < m_{0}$ are used to test the mean using a Poisson distribution.
To find the critical region when a 1% level of significance is used, you find the largest value $c$ such that $P (X \leq c) \leq 0.01$ , where $X ~ Po (m_{0})$ .
This means $P (X \leq c + 1) > 0.01$ , where $X ~ Po (m_{0})$ .
True or False?
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m > m_{0}$ are used to test the mean using a Poisson distribution.
A random interval is observed and the number of occurrences, $x$ , is recorded.
The critical value is the smallest value, $c$ , such that $P (X \geq c)$ , where $X ~ Po (m_{0})$ is greater than the significance level.
False.
The hypotheses $H_{0} : m = m_{0}$ and $H_{1} : m > m_{0}$ are used to test the mean using a Poisson distribution.
A random interval is observed and the number of occurrences, $x$ , is recorded.
The critical value is the smallest value, $c$ , such that $P (X \geq c)$ , where $X ~ Po (m_{0})$ is less than the significance level.
The mean of $X ~ Po (m)$ is being tested using the hypotheses $H_{0} : m = 2.5$ and $H_{1} : m > 2.5$ at a 5% level of significance.
What is the critical value if $P (X \geq 6) = 0.04202$ and $P (X \geq 5) = 0.10882$ , where $X ~ Po (2.5)$ ?
The mean of $X ~ Po (m)$ is being tested using the hypotheses $H_{0} : m = 2.5$ and $H_{1} : m > 2.5$ at a 5% level of significance.
If $P (X \geq 6) = 0.04202$ and $P (X \geq 5) = 0.10882$ , where $X ~ Po (2.5)$ , the critical value is 6.
The mean of $X ~ Po (m)$ is being tested using the hypotheses $H_{0} : m = 17$ and $H_{1} : m < 17$ at a 5% level of significance.
Given that $P (X \leq 9) = 0.02612$ , where $X ~ Po (17)$ , which probability should you check next if you are trying to find the critical region?
The mean of $X ~ Po (m)$ is being tested using the hypotheses $H_{0} : m = 17$ and $H_{1} : m < 17$ at a 5% level of significance.
Given that $P (X \leq 9) = 0.02612$ , you check $P (X \leq 10)$ next if you are trying to find the critical region, where $X ~ Po (17)$ .
This is because the first probability is less than the significance level so you should try to increase the size of the interval.
True or False?
A t-test can be used to test whether there is a linear correlation between two normally distributed variables.
True.
A t-test can be used to test whether there is a linear correlation between two normally distributed variables.
In the context of hypothesis testing, what is denoted by $ρ$ ?
$ρ$ denotes the population product moment correlation coefficient between two normally distributed variables.
What does the null hypothesis look like for a test for linear correlation?
The null hypothesis for a test for linear correlation is $H_{0} : ρ = 0$ .
True or False?
The alternative hypothesis for a test for linear correlation is always $H_{1} : ρ \neq 0$ .
False.
The alternative hypothesis for a test for linear correlation is not always $H_{1} : ρ \neq 0$ . It could also be $H_{1} : ρ < 0$ or $H_{1} : ρ > 0$ .
What would the alternative hypothesis be if you were testing for a positive linear correlation?
The alternative hypothesis would be $H_{1} : ρ > 0$ if you were testing for a positive linear correlation.
What is the difference between a one-tailed test and a two-tailed test for linear correlation?
A one-tailed test for linear correlation tests specifically for a particular type of correlation (positive or negative). Whereas, a two-tailed test just tests for any linear correlation.
A two-tailed test is used when you do not know what type of correlation there might be.
What would the conclusion be if the p-value for a two-tailed test for linear correlation is less than the significance level?
If the p-value for a two-tailed test for linear correlation is less than the significance level, then there is sufficient evidence to suggest that there is a linear correlation between the two variables.
Note that the test does not determine whether it is a positive or negative correlation.
What would the conclusion be if the p-value for a test for positive linear correlation is greater than the significance level?
If the p-value for a test for positive linear correlation is greater than the significance level, then there is insufficient evidence to suggest that there is a positive linear correlation between the two variables.
What is a Type I error?
A Type I error occurs if the null hypothesis is rejected when it is true.
What is a Type II error?
A Type I error occurs if the null hypothesis is not rejected when it is not true.
True or False?
The probability of a Type I error is always equal to the stated significance level.
False.
The probability of a Type I error is always less than or equal to the stated significance level.
For continuous distributions, the probability is equal to the stated significance level.
True or False?
The critical region will maximise the probability of a Type I error while keeping it less than or equal to the stated significance level.
True.
The critical region will maximise the probability of a Type I error while keeping it less than or equal to the stated significance level.
If the null hypothesis is not rejected, what type of error could have been made?
If the null hypothesis is not rejected, then a Type II error could have been made.
True or False?
$P (Type II error) = 1 - P (Type I error)$ .
False.
In general, $P (Type II error) \neq 1 - P (Type I error)$ .
True or False?
The hypotheses $H_{0} : μ = 100$ and $H_{1} : μ < 100$ are used.
The critical region is $\bar{X} < 98$ .
If the true mean is $μ = 99$ , then the probability of a Type II error is equal to $P (\bar{X} < 98 | μ = 99)$ .
False.
The hypotheses $H_{0} : μ = 100$ and $H_{1} : μ < 100$ are used.
The critical region is $\bar{X} < 98$ .
If the true mean is $μ = 99$ , then the probability of a Type II error is equal to $P (\bar{X} > 98 | μ = 99)$ .
You find the probability of obtaining a value that is not in the critical region.
The hypotheses $H_{0} : μ = 100$ and $H_{1} : μ < 100$ are used.
The critical region is $\bar{X} < 98$ .
How would you find the probability of a Type I error?
The hypotheses $H_{0} : μ = 100$ and $H_{1} : μ < 100$ are used.
The critical region is $\bar{X} < 98$ .
The probability of a Type I error is equal to $P (\bar{X} < 98 | μ = 100) .$
The hypotheses $H_{0} : μ = 100$ and $H_{1} : μ \neq 100$ are used.
The critical regions are $\bar{X} < 99$ and $\bar{X} > 101$ .
If the true value of the mean is $μ = 99.5$ , how would you find the probability of a Type II error?
The hypotheses $H_{0} : μ = 100$ and $H_{1} : μ \neq 100$ are used.
The critical regions are $\bar{X} < 99$ and $\bar{X} > 101$ .
If the true value of the mean is $μ = 99.5$ , then the probability of a Type II error is equal to $P (99 < \bar{X} < 101 | μ = 99.5) .$
The hypotheses $H_{0} : p = 0.3$ and $H_{1} : p > 0.3$ are used.
The critical region is $X \geq 17$ .
The true value of the proportion is $p = 0.35$ .
How would you find the probability of a Type II error?
The hypotheses $H_{0} : p = 0.3$ and $H_{1} : p > 0.3$ are used.
The critical region is $X \geq 17$ .
The true value of the proportion is $p = 0.35$ .
The probability of a Type II error is equal to $P (X \leq 16 | p = 0.35)$ .
The hypotheses $H_{0} : m = 7.5$ and $H_{1} : m < 7.5$ are used.
When a 5% significance level is used, the critical region is $X \leq 2$ .
Write an inequality for the probability of a Type I error.
The hypotheses $H_{0} : m = 7.5$ and $H_{1} : m < 7.5$ are used.
When a 5% significance level is used, the critical region is $X \leq 2$ .
An inequality for the probability of a Type I error would be $P (Type I error) \leq 0.05$ .
The probability of which type of error usually decreases when the significance level is increased?
The probability of a Type II error usually decreases when the significance level is increased.

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Number Toolkit

Exponentials & Logs

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Further Complex Numbers

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2. Functions

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Further Functions & Graphs

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Functions Toolkit

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Hypothesis Testing using the Chi-squared Distribution

Hypothesis Testing for Population Parameters

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