Confidence intervals: notes and practice questions
- Confidence Intervals for the Mean (with Sample Mean Distribution & Central Limit Theorem) are HL syllabus.
- The sample mean () is used as a point estimate for the population mean ().
- A Confidence Interval for the Mean is an interval where the exact population mean () is likely to lie.
- The width of a confidence interval is the total range of values it contains.
- Increasing the confidence level increases the interval width.
- Decreasing the confidence level decreases the interval width.
- Increasing the sample size decreases the interval width.
- Decreasing the sample size increases the interval width.
- If a claimed value for is within the confidence interval, there is not enough evidence to reject the claim.
- If a claimed value for is outside the confidence interval, there is sufficient evidence to reject the claim.
How it is examined
Produce the interval on the GDC, then interpret it. The interpretation mark is the harder one and the TOK note above is effectively the model answer: two overlapping intervals do not support a claim of difference. Choosing normal against is a decision mark and depends only on whether is known.
Confidence intervals for the mean of a normal population.
Linking questions
- Other contexts: forecasting, and attaching value to claims and predictions.
- Links to other subjects: analysis of data from field studies (sciences and individuals and societies).
- TOK: mathematics and the world. Claiming brand A is "better" on average than brand B can mean very little if there is a large overlap between the confidence intervals of the two means.
Practice questions
5 questions · 5 mediumQuestion 1
MediumPaper 1 · calculator6 marksA materials scientist is developing a new synthetic fiber and claims its average tensile strength is 120 N. To test this claim, she takes a random sample of 25 fiber segments.
Given that the tensile strength of an individual fiber segment is newtons, the scientist found that, from her 25 samples, and .
(a) Find an unbiased estimate for the mean tensile strength () of the fiber.
(b) Use the formula to determine an unbiased estimate for the variance of the tensile strength of the fiber.
(c) Find a 95% confidence interval for . You may assume that all conditions for a confidence interval have been met.
(d) Suggest, with justification, a valid conclusion that the materials scientist could make regarding her claim.
Recall the formula for the sample mean, which is an unbiased estimate for the population mean.
Carefully substitute the given values into the formula for the unbiased sample variance. Remember to use in the denominator.
You will need the sample mean and the unbiased standard deviation (square root of the variance) from parts (a) and (b). For a 95% confidence interval with a small sample size and unknown population standard deviation, use the t-distribution. You can use your GDC to find the critical t-value or the entire confidence interval.
Compare the claimed average tensile strength (120 N) with the 95% confidence interval you calculated in part (c). If the claimed value falls outside the interval, what does that imply?
Question 2
MediumPaper 1 · calculator9 marks(a) A quality control inspector at a beverage company takes a random sample of eight bottles of orange juice from a production line. The measured volumes, in ml, are:
298.5, 301.2, 299.1, 300.8, 298.9, 301.5, 299.7, 300.3
(i) Find an unbiased estimate for the mean volume of orange juice in a bottle from this production line.
(ii) Calculate a 95% confidence interval for the population mean volume. Give your answer to four significant figures.
(b) State one assumption you have made in order for your interval to be valid.
(c) The label on each bottle states: "Volume: 300 ml".
Using your answer to part (a)(ii), briefly comment on the claim on the label.
To find an unbiased estimate for the population mean, you should calculate the sample mean of the given data.
You will need to calculate the sample standard deviation (unbiased estimate) and use the t-distribution for the confidence interval, as the population standard deviation is unknown and the sample size is small.
Consider the underlying distribution of the data when constructing a t-interval for the mean.
Check if the claimed value falls within the calculated confidence interval. What does this imply about the claim?
Question 3
MediumPaper 2 · calculator7 marks(a) State the Central Limit Theorem, including the conditions under which it applies.
(b) A logistics company is investigating the average commute time for its employees. It is known that the commute times have a population variance of minutes. A random sample of employees reported their daily commute times, and the sample mean was minutes.
Calculate the confidence interval for the population mean commute time, .
Recall the conditions regarding sample size and the distribution of the sample mean when the population distribution is unknown or not normal.
Use the formula for a confidence interval for the population mean when the population standard deviation is known: . Remember to find the correct -critical value for a confidence level.
Question 4
MediumPaper 1 · calculator6 marksA food scientist is evaluating a new genetically modified apple variety, 'Crimson Crisp', known for its potential to grow larger. A random sample of 8 apples from the first harvest was weighed. The sample mean weight was grams, and the sample standard deviation of this sample, , was grams.
(a) Find an unbiased estimate of the population variance for the weight of 'Crimson Crisp' apples.
(b) Assuming that the weights of 'Crimson Crisp' apples are normally distributed, find a 95% confidence interval for the true mean weight of this apple variety.
(c) Using your answer to part (b), determine if it is plausible that the true mean weight of 'Crimson Crisp' apples could be grams. Justify your answer.
Recall the formula for the unbiased estimate of population variance, , which uses the sample standard deviation and the sample size . The formula is .
Since the population standard deviation is unknown and the sample size is small, you should use a -interval. Remember to use the unbiased estimate of the standard deviation () or ensure your GDC is set up to do so when calculating the interval.
Check if the given mean value falls within the confidence interval you calculated in part (b). If it does, it is plausible; otherwise, it is not.
Question 5
MediumPaper 1 · calculator5 marks(a) A quality control manager at a beverage factory takes a random sample of bottles to check the volume of juice. The sample mean volume is ml and the sample standard deviation () is ml.
Calculate the unbiased estimate of the population standard deviation, , for this sample.
(b) Find a 95% confidence interval for the population mean volume, giving your answer to 3 significant figures.
(c) The bottles are labelled as containing ml of juice. Comment on this claim with reference to your answer in part (b).
Remember the relationship between the sample standard deviation () and the unbiased estimate of the population standard deviation (). The formula is .
Use the t-distribution for the confidence interval since the population standard deviation is unknown and the sample size is small. You will need the sample mean, the unbiased standard deviation from part (a), the sample size, and the degrees of freedom (). A GDC's t-interval function is appropriate.
Consider whether the claimed volume of ml falls within the 95% confidence interval you calculated in part (b). If it does not, what does that imply about the claim?
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Where marks are lost
- Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three. Common wherever a GDC's full decimal display gets copied straight down.
- Rounding an intermediate value and then using it in a later part. Costs a mark every time, and AI's multi-part modelling questions give it more chances to happen than AA's shorter, more self-contained ones.
- Writing the answer and nothing else, where the mark scheme has an explicit M1 rather than an implied one. A bare answer cannot score full marks there.
