Confidence Interval Practice Questions with Answers

Concept Explanation

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter with a specific level of certainty. In statistics, we rarely know the exact population mean or proportion, so we use a confidence interval to provide an estimated range based on our data. The most common levels of confidence are 95%, 90%, and 99%, which represent how frequently the calculated interval would contain the true parameter if we were to repeat the sampling process many times. For more complex calculations involving molecular data, you might find our Mass Spectrometry Practice Questions with Answers helpful for understanding data precision.

The general formula for a confidence interval is: Point Estimate ± (Critical Value × Standard Error).

Key Components

• Point Estimate: The sample statistic (like the sample mean $\bar{x}$ or sample proportion $\hat{p}$) used as the best guess for the population parameter.

• Critical Value: A multiplier (like $z^*$ or $t^*$) based on the desired confidence level and the sampling distribution. For large samples (n > 30) or known population standard deviation, we use the Z-distribution. For smaller samples with unknown standard deviation, we use the Student's t-distribution.

• Standard Error (SE): The standard deviation of the sampling distribution of the statistic. For a mean, $SE = \sigma / \sqrt{n}$.

• Margin of Error (MOE): The product of the critical value and the standard error. It represents the maximum expected difference between the sample statistic and the true population parameter.

According to the Khan Academy, the width of the interval is influenced by the sample size and the confidence level. Increasing the sample size decreases the margin of error, making the interval narrower and more precise. Conversely, increasing the confidence level (e.g., from 95% to 99%) increases the margin of error, making the interval wider.

Solved Examples

Below are 3 fully worked examples illustrating how to calculate a confidence interval for means and proportions.

Example 1: Confidence Interval for a Mean (Z-distribution)

A researcher wants to estimate the average height of a specific plant species. A random sample of 50 plants has a mean height of 15 cm. The population standard deviation is known to be 2 cm. Calculate a 95% confidence interval.

1. Identify the knowns: $\bar{x} = 15$, $\sigma = 2$, $n = 50$, Confidence Level = 95%.

2. Find the critical value: For 95% confidence, $z^* = 1.96$.

3. Calculate the Standard Error: $SE = 2 / \sqrt{50} \approx 0.283$.

4. Calculate the Margin of Error: $MOE = 1.96 \times 0.283 = 0.555$.

5. Determine the interval: $15 \pm 0.555 = (14.445, 15.555)$.

Example 2: Confidence Interval for a Proportion

In a survey of 400 people, 160 stated they prefer coffee over tea. Find the 90% confidence interval for the population proportion of coffee drinkers.

1. Calculate the sample proportion: $\hat{p} = 160 / 400 = 0.4$.

2. Identify the critical value: For 90% confidence, $z^* = 1.645$.

3. Calculate the Standard Error: $SE = \sqrt{(0.4 \times 0.6) / 400} = \sqrt{0.0006} = 0.0245$.

4. Calculate the Margin of Error: $MOE = 1.645 \times 0.0245 = 0.0403$.

5. Determine the interval: $0.4 \pm 0.0403 = (0.3597, 0.4403)$.

Example 3: Small Sample Mean (T-distribution)

A small batch of 10 chemical samples was tested for purity. The sample mean was 98.2% with a sample standard deviation of 0.5%. Calculate a 95% confidence interval.

1. Identify the knowns: $\bar{x} = 98.2$, $s = 0.5$, $n = 10$, $df = 9$.

2. Find the critical value: Using a t-table for $df=9$ at 95%, $t^* = 2.262$.

3. Calculate the Standard Error: $SE = 0.5 / \sqrt{10} = 0.158$.

4. Calculate the Margin of Error: $MOE = 2.262 \times 0.158 = 0.357$.

5. Determine the interval: $98.2 \pm 0.357 = (97.843, 98.557)$.

Practice Questions

1. A sample of 100 light bulbs has an average lifespan of 1,200 hours. The population standard deviation is 50 hours. Calculate the 95% confidence interval for the mean lifespan.

2. A poll of 1,000 voters shows that 520 support a new tax law. Find the 99% confidence interval for the true proportion of voters who support the law.

3. A researcher measures the weight of 16 laboratory mice. The mean weight is 25 grams with a sample standard deviation of 3 grams. Calculate a 90% confidence interval for the mean weight (use t-distribution).

4. A fitness tracker company claims their device is accurate. A test on 64 users shows a mean error of 2 steps per minute with a population standard deviation of 0.8 steps. Find the 98% confidence interval.

5. If you want to reduce the margin of error by half in a mean estimation problem while keeping the confidence level the same, by what factor must you increase the sample size?

6. In a clinical trial, 15 out of 200 patients reported side effects. Calculate the 95% confidence interval for the proportion of patients experiencing side effects.

7. You are given a 95% confidence interval of (45, 55). What is the point estimate and the margin of error?

8. A student wants to estimate the mean score on a standardized test. With a known population standard deviation of 15, what sample size is needed to achieve a margin of error of 2 points at a 95% confidence level?

9. A researcher finds a 95% confidence interval for a mean to be (10.2, 14.8). If they increase the confidence level to 99% using the same data, will the interval become wider or narrower?

10. Calculate the 95% confidence interval for a mean where $n=25$, $\bar{x}=100$, and $s=10$ (use t-distribution).

Answers & Explanations

1. (1190.2, 1209.8): $z^* = 1.96$, $SE = 50 / \sqrt{100} = 5$. $MOE = 1.96 \times 5 = 9.8$. Interval = $1200 \pm 9.8$.

2. (0.479, 0.561): $\hat{p} = 0.52$, $z^* = 2.576$. $SE = \sqrt{(0.52 \times 0.48) / 1000} \approx 0.0158$. $MOE = 2.576 \times 0.0158 \approx 0.041$.

3. (23.72, 26.28): $df = 15$, $t^* = 1.753$. $SE = 3 / \sqrt{16} = 0.75$. $MOE = 1.753 \times 0.75 = 1.315$.

4. (1.767, 2.233): $z^*$ for 98% is 2.33. $SE = 0.8 / \sqrt{64} = 0.1$. $MOE = 2.33 \times 0.1 = 0.233$.

5. Factor of 4: Since the margin of error involves $1/\sqrt{n}$, reducing the error by half requires increasing $n$ by the square of that change ($2^2 = 4$).

6. (0.038, 0.112): $\hat{p} = 15/200 = 0.075$. $z^* = 1.96$. $SE = \sqrt{(0.075 \times 0.925) / 200} \approx 0.0186$. $MOE = 1.96 \times 0.0186 \approx 0.0365$.

7. Point Estimate = 50, MOE = 5: The point estimate is the midpoint $(45+55)/2 = 50$. The MOE is the distance from the midpoint to either bound.

8. 217: $n = (z^* \times \sigma / MOE)^2 = (1.96 \times 15 / 2)^2 = (14.7)^2 = 216.09$. Always round up for sample size.

9. Wider: Increasing the confidence level requires a larger critical value ($z^*$), which increases the margin of error and makes the interval wider. For more on how variables affect outcomes, check our Medium Reaction Mechanism Practice Questions.

10. (95.87, 104.13): $df = 24$, $t^* = 2.064$. $SE = 10 / \sqrt{25} = 2$. $MOE = 2.064 \times 2 = 4.128$.

Quick Quiz

Frequently Asked Questions

What does a 95% confidence interval actually mean?

It means that if we were to take many random samples and calculate a confidence interval for each, approximately 95% of those intervals would contain the true population parameter. It is a statement about the reliability of the estimation procedure over many repetitions.

Does a wider confidence interval mean better results?

No, a wider interval usually indicates less precision, often due to a smaller sample size or a higher required confidence level. While you are "more sure" the parameter is inside, the range is less specific and therefore often less useful for decision-making.

How do I choose between a Z-test and a T-test for intervals?

Use a Z-test when the population standard deviation is known or the sample size is large (n > 30). Use a T-test when the population standard deviation is unknown and you must use the sample standard deviation, especially with smaller samples.

Can a confidence interval be 100%?

In practical statistics, a 100% confidence interval would have to cover all possible values (from negative infinity to positive infinity for a mean), making it useless. We accept a small risk of being wrong (alpha) to provide a meaningful, finite range. If you are interested in definite structures, see our Easy Naming Organic Compounds (IUPAC) Practice Questions.

What is the relationship between margin of error and confidence level?

The margin of error is directly proportional to the confidence level; as you increase the confidence level, the critical value increases, which in turn increases the margin of error. This results in a wider interval to ensure a higher probability of capturing the parameter.

How does population variance affect the confidence interval?

Higher population variance leads to a larger standard error, which increases the margin of error and widens the confidence interval. Essentially, more "noise" or spread in the population makes it harder to pinpoint the exact mean.

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