Hypothesis Testing (DP IB Maths: AI HL)

A new technology company, TechBright, has developed a battery that they claim has a longer lifespan than the product sold by its main competitor, Elektrik. A survey has been completed recording the battery life, in hours, of 12 batteries from each company. The results are shown in the table below.

State the null and alternative hypotheses.

Perform a t-test at the 5% significance level.

State the conclusion of this test, giving a reason for your answer.

If the test had instead been conducted at a significance level of 10%, state the conclusion for the t-test, giving a reason for your answer.

Determine if TechBright is correct in claiming that their batteries have a longer lifespan than those created by Elektrik. Justify your answer.

A football coach wants to know if the number of hours spent training by a team in the week before a match affects the outcome of the game.

Data is collected for 100 matches, with the number of hours spent training during the preceding week the match, h, recorded alongside the result of win, lose or draw.  Some of this data is included in the following table:

	Win	Lose	Draw	Total
	4		11	31
		19	12
	18	4	1	23
Total	37	39		100

Complete the table of observed data.

State the null hypothesis.

Calculate the expected number of losses for a team that trains between 16 and 24 hours in the week before a match if training time and match results are independent of each other.

Find the number of degrees of freedom.

Calculate the test statistic for this data, testing at the 10% significance level.

The critical value is 7.779. State the conclusions obtained and justify your answer.

A spinner used on a game show is divided into 6 equal parts coloured red, yellow, green, blue, purple and orange.  There are reports that the spinner is not fair so an experiment is conducted to determine if this is the case.  The spinner is spun 720 times.

Calculate the number of times you would expect the spinner to land on red.

The results of the experiment are shown in the table below:

Write down the statistical test that can be performed to determine if the spinner is fair by comparing the observed results with the expected results.

Write down the null and alternative hypotheses.

Perform the test, using technology, and find the p-value.

If the test is performed at the 1% significance level, state the conclusion of the test. Give a reason for your answer.

Comment on the suitability of the significance level. Justify your answer.

A second spinner undergoes the same experiment, which results in a p-value of 0.082.

Explain how the difference in the p-values could be used to comment upon the fairness of the two spinners.

150 people are surveyed as part of an investigation into the relationship between gender and sport.  The researcher asks people to choose their preferred sport for viewing from a choice of football, basketball, tennis, gymnastics and swimming.

The results of the survey are shown in the table below.

A chi-squared test is performed on the data at a significance level of 5%.

State the null and alternative hypotheses.

Using the table of critical values for a significance level of 5%, given below, write down the appropriate critical value.

Calculate the  statistic.

State the conclusion to the test, giving a reason for your answer.

A gardener believes that sunflowers situated in areas of full sun will grow taller than sunflowers planted in partial sun.  To investigate this, the gardener measures the heights, in inches, of sunflowers growing under both conditions.  The results are shown in the table below.

A t-test is performed at a significance level of 5%.

State the null and alternative hypotheses.

Calculate the p-value for the data.

Comment on the results of the test.

Comment on the result of the test if the significance level had instead been 1%.

If the experiment were to be repeated, suggest one step that could be taken to increase the validity of the results.

A student is investigating the relationship between a country’s GDP and its literacy rate for his IA.  He has classified 140 countries into low, medium and high GDP and their literacy rates into low, medium and high.  His results are displayed in the table below.

Perform a chi-squared test on the data at a 5% significance level to test the hypothesis that the literacy rate of a country is dependent on its GDP.  The critical value is 9.488.

State the null and alternative hypotheses.

Write down the number of degrees of freedom.

Calculate the p-value and the statistic. You should justify any conclusions found.

Describe one possible issue with the way the data is presented, that might make it difficult to interpret the validity or precise implications of the test’s conclusions.

A computer game has 5 levels.  At the end of each level a magic star may appear, doubling your score from that level. The game is played through all levels 200 times by a focus group.  The number of times in total that the magic star appears for each game of 5 levels is recorded, and the results for all 200 games are summarised in the table below.

The game developer wants to investigate whether the results can be modelled using the binomial distribution .

State the null and alternative hypotheses.

Draw a table of expected frequencies.

A chi-squared goodness of fit test is performed at the 10% significance level. State the  statistic.

The critical value for the test is 9.236. Comment on the results of the test, justifying your answer.

In healthy adults, systolic blood pressure is normally distributed with a mean of 112 mmHg and standard deviation of 10 mmHg.  A group of 200 patients take part in a clinical trial and their systolic blood pressure, , is measured and recorded below.

Systolic blood pressure (mmHg)	Frequency
	6
	59
	65
	63
	7

A chi-squared goodness of fit test at a significance level of 5% is used to determine if the sample of patients is representative of the general population in terms of their blood pressure. 

State the null and alternative hypotheses.

Complete the table of expected frequencies below, giving the frequencies to 4 decimal places.

Systolic blood pressure (mmHg)	Expected frequency
	39.4796
	75.1896

State the number of degrees of freedom.

Calculate the p-value and comment on the results of the test.

The tech company Flooglesoft has recently lost some key employees to its main competitor.  As a reaction to this the company is interested to see whether the wages of its employees can be modelled by a normal distribution with standard deviation $400.  The data for the monthly wages of Flooglesoft’s employees are summarised in the table below.  

Monthly wage, w ($)	Frequency
	1
	8
	28
	76
	99
	89
	44
	15

Given that , find the unbiased estimate for the population mean.

A  goodness of fit test is to be conducted at a significance level of 10% to test whether the data can be modelled by a normal distribution with standard deviation $400.

State the null and alternative hypotheses.

Find the values of a, b and c in the table below. Give your answers to 4 decimal places.

Monthly wage ($)	Expected frequency
	a
	7.1173
	29.8439
	73.0396
	b
	87.3498
	c
	14.4213

 

Explain why there are 5 degrees of freedom.

The critical value is 9.236. Calculate the statistic and comment on the conclusion of the test.  Justify your answer.

Leander works for a manufacturing company. Each shift lasts four hours and during this time she records the number of times a machine breaks down. The data from 160 shifts are shown in the table below.

Leander believes that the number of breakdowns during a four-hour shift can be modelled by a Poisson distribution. She decides to test her belief using a  goodness of fit test using a 5% level of significance.

Write down suitable null and alternative hypotheses to test Leander’s belief.

Calculate an estimate for the mean number of breakdowns during a four-hour shift.

Using the estimated mean, find the values of a, b and c in the table below. Give your answers to 3 decimal places.

Write down the number of degrees of freedom.

Determine whether the conclusion from the test supports Leander’s belief. Justify your answer.

An IB student is investigating the study habits of her peers. She decides to conduct a study to find out if the time that a student prefers to study is affected by their gender. She selects 20 boys and 20 girls at random from her year group and asks them to pick their preferred time of study from the options of morning, afternoon or evening.

State the type of sampling used in her investigation.

State the null and alternative hypotheses.

The results of the survey are listed below.

Create a contingency table of observed values.

Show that if the preferred study time and gender are independent then the expected number of  female students that prefer to study in the morning is 6.5.

The critical value for a chi-squared test performed at a significance level of 5% is 5.991.

Using technology, calculate the statistic and comment on the conclusions of the test in the context of the question, giving a reason for your answer.

A group of university researchers are concerned that pollution from industrial activity is having an effect on the local river system.  In order to investigate whether runoff from local factories is affecting the water quality, samples were taken from next to the factories as well as from upstream.  The data from both locations is compared to test whether the average pH is higher next to the factories than it is upstream. The test is conducted at a significance level of 1%.  The pH of the samples is recorded in the table below.

State the type of distribution of the data.

State the null and alternative hypotheses.

Calculate the p-value and state any conclusions of the test, giving a reason for your answer.

Interpret the implications of the conclusion to the test for the researchers.

It is thought that agricultural runoff will also negatively affect the water quality.  Samples have been taken near a number of farms and compared to the upstream samples at the same significance level. The p-value for this test is 0.0082.

Comment on the result of this test and compare it to the initial test. Justify your answer.

A cat shelter looks after 328 cats that have been abandoned or neglected.  As part of the initial health check of a new cat, its weight, W, in pounds is measured and recorded.  The table below collects together the data for the cat weights.

Weight	Frequency
	17
	91
	141
	72
	7

The weight distribution of the general cat population can be described by

W~N(10.4, 0.9²).  The shelter wants to see if the weight distribution of the cats in their care fits that of the general cat population.

A chi-squared goodness of fit test is to be performed at the 10% significance level.

Write down the null and alternative hypotheses.

Draw a new table showing the expected frequencies if the shelter cat population were to match the general cat population weight distribution.

Calculate the p-value.

Comment on the results of the test and the conclusions that can be drawn, giving a reason for your answer.

A lottery machine contains a set of coloured balls: three red, five yellow, two green and two blue.  When a button is pressed a ball is selected at random.  After noting its colour, the same ball is returned to the machine.

Calculate the probability of the machine selecting a yellow ball at random.

The machine is to be tested for bias and so the procedure of selecting a ball at random is repeated 180 times and the results are recorded in the table below.

Draw a table showing the expected frequencies for the test.

Perform a goodness of fit test at a significance level of 5% and comment on your results, giving a reason for your answer.

Approximately 1.6 million pairs of twins are born each year world-wide

Calculate the probability that both  babies in any random  set of twins are male .State any assumptions you have made.

Complete the probability table for the number of male babies born in a given set of twins.

In one particular hospital, there are 236 sets of twins born in one year.

Complete the expected frequency table.

The actual number of male babies born as part of a set of twins in this hospital during the year is shown in the table below.

Perform a chi-squared goodness of fit test, given that the critical value for the test is is 5.991. Comment on the results of the test, giving a reason for your answer.

A group of 14 students are studying reaction times in their science lesson.  They use a computer program to measure the time taken, in milliseconds, for a button to be pressed after a green light is shown.  One of the students has compared the group’s results to those of the expected normal distribution of reaction times and completed a chi-squared goodness of fit test at a significance level of 10%.  The calculated p-value is 0.130.

Comment on her results, justifying your answer.

The student noted that in the first test, the dominant hand was used to press the button. It was decided to repeat the experiment using the non-dominant hand.  The calculated p-value for this test, which was also conducted with a significance level of 10%, is 0.0854.

Compare the results of both tests. Justify your answer.

The data from both tests are then compared using a two-tailed t-test at a significance level of 10%.  The p-value for this test is 0.0667.

Explain the result of this test.

A study has noted that the top 20% of phone users spend more than 4.5 hours per day on their phones.  An independent health charity wishes to investigate whether heavy phone usage is linked to the age of the user.  A survey of 85 people is commissioned where the age group (child, teenager or adult) and the average number of hours, h, that are spent on the phone each day are noted.

The results are displayed in the table below.

	Low usage	Medium usage	High usage	Total
Child	15	8	3	26
Teenager	6	18	8	32
Adult	9	12	6	27
Total	30	38	17	85

Draw a contingency table of expected values.

Perform a chi-squared test and comment on the results. You should state the null and alternative hypotheses and comment on any conclusions found.

Select the appropriate critical value from the table below.

State two steps that could be taken to increase the validity of the conclusions drawn from the test.

Gorr and Greer are butchers and purchase beef from two different suppliers: Beefthor and Zeusbeef. Gorr and Greer use the beef to make 150 gram burgers. Gorr suspects that the mean amount of fat in the beef is different between the two suppliers. Gorr measures the amount of fat, in grams, in a sample of 150 gram burgers using beef from the two suppliers. The data is shown in the table below.

Gorr uses a pooled two-sample t-test to test his suspicion using a 5% significance level.

State two assumptions that Gorr has made about the distribution of the amount of fat in the burgers using beef from each of the two suppliers.

Perform the hypothesis test to test Gorr’s suspicion. State the hypotheses clearly and justify your conclusion.

Greer suspects that the beef from Zeusbeef has more fat than the beef from Beefthor. Use the data from Gorr’s sample to test Greer’s suspicion using a 5% significance level. State the hypotheses clearly and justify your conclusion.

Explain why Gorr and Greer’s tests have different conclusions.

Gorr and Greer use another sample to test their different suspicions using the same significance level. There is sufficient evidence from this sample to support Gorr’s suspicion. Explain whether Greer’s suspicion will also be supported using this sample.

Melanie is playing an online game which has two digital four-sided dice numbered 1 to 4. The pair of digital dice are rolled together three times and points are scored. High points are scored for rolling doubles this is when both dice land on the same number. 

The following table shows the distribution of the number of doubles scored in each set of three throws when the Melanie plays the game 400 times. 

Melanie suspects that the data can be modelled using a binomial distribution with the probability of rolling a double being 0.25. A  goodness of fit test is to be used with a 5% significance level to test Melanie’s suspicion. 

Perform the test to show that the data can not be modelled accurately by the binomial distribution B(3, 0.25).

Melanie still suspects that the data can be modelled using a binomial distribution. A  goodness of fit test is to be used with a 5% significance level to test Melanie’s suspicion.

Assuming that each dice roll is independent, use the table of results to estimate the probability of rolling doubles with the pair of dice.

Gordon takes the bus to work at the same time each day and he records the time, in minutes, of each journey. He collates his data in the table below.

Time (t minutes)	Frequency
	6
	11
	17
	21
	14
	1

Gordon wants to use a  goodness of fit test with a 10% significance level to see whether the journey times can be modelled by a normal distribution.

State the null and alternative hypotheses.

The mean time of the sample is 25.3 minutes and the standard deviation for the sample is 7.38 minutes.

Write down the unbiased estimate for the population mean time and find the unbiased estimate for the population variance.