5-3 Estimation and Confidence Intervals: practice questions
Unit 5 — Normal Distribution · Statistics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 90 questions on this lesson, each with an explanation.
A constant numerical value that characterizes a population and is often unknown, such as the population mean μ, is called ……
Athe critical value
Ba confidence interval
Ca statistic
Da parameter
Show answer and explanation
Correct answer: (D) a parameter
Explanation: Step 1: A parameter is a constant numerical value that characterizes a population and is often unknown, such as the population mean μ.
Step 2: It is estimated by a statistic calculated from the sample, such as the sample mean X̄.
∴ it is called a parameter (a statistic is a trap: it comes from the sample, not the population).
Question 2Easy
If the upper limit of the confidence interval for a population mean is 48.5 and the estimation error is 2.5, then the arithmetic mean of the sample = ……
A47.25
B51
C43.5
D46
Show answer and explanation
Correct answer: (D) 46
Explanation: Step 1: Upper limit = X̄ + E.
Step 2: 48.5 = X̄ + 2.5, so X̄ = 48.5 − 2.5 = 46.
∴ X̄ = 46 (51 is a trap: adding E instead of subtracting it; 43.5 subtracts 2E and is the lower limit).
Question 3Easy
A sample of size 36 is taken from a normal population with standard deviation σ = 12. Given that the critical value for the 99% confidence level is Zα/2 = 2.58, the estimation error E at the 99% confidence level = ……
A0.86
B5.16
C3.92
D10.32
Show answer and explanation
Correct answer: (B) 5.16
Explanation: Step 1: E = (σ/√n) × Zα/2.
Step 2: E = (12/√36) × 2.58 = 2 × 2.58 = 5.16.
∴ E = 5.16 (3.92 is a trap: it uses 1.96, the value for 95%; 0.86 divides by 36 instead of 6; 10.32 is 2E).
Question 4Medium
A sample of 100 customers was taken. The average bill value was 250 L.E. and the standard deviation was 20 L.E. The 95% confidence interval for the average bill value (critical value Zα/2 = 1.96) is ……
A]249.608, 250.392[
B]246.08, 253.92[
C]242.16, 257.84[
D]210.8, 289.2[
Show answer and explanation
Correct answer: (B) ]246.08, 253.92[
Explanation: Step 1: n = 100 > 30, so we take σ = 20.
Step 2: E = (20/√100) × 1.96 = 2 × 1.96 = 3.92.
Step 3: ]X̄ − E, X̄ + E[ = ]250 − 3.92, 250 + 3.92[ = ]246.08, 253.92[.
∴ ]246.08, 253.92[ (dividing by 100 instead of 10 gives E = 0.392; forgetting √n gives E = 39.2; using 2E = 7.84 gives ]242.16, 257.84[).
Question 5Medium
If the sample size and the confidence level are kept fixed, then for a larger population standard deviation σ, the length of the confidence interval for the population mean ……
Ais smaller
Bis greater
Cdoes not change
Dis halved
Show answer and explanation
Correct answer: (B) is greater
Explanation: Step 1: E = (σ/√n) × Zα/2 is directly proportional to σ.
Step 2: A larger σ gives a larger E, and the length of the interval is 2E.
∴ the length of the interval is greater.
Question 6Medium
If the critical value Zα/2 = 1.75 and P(0 ≤ Z ≤ 1.75) = 0.4599, then the confidence level ≈ ……
A8%
B46%
C92%
D96%
Show answer and explanation
Correct answer: (C) 92%
Explanation: Step 1: (1 − α)/2 = 0.4599.
Step 2: 1 − α = 2 × 0.4599 = 0.9198 ≈ 0.92.
∴ the confidence level ≈ 92% (46% is a trap: it is the area before multiplying by 2; 8% is α).
Question 7Medium
A confidence interval for a population mean is calculated from a sample at the 95% confidence level, and then from the same sample at the 99% confidence level. Compared with the first interval, the second interval is ……
Alonger
Bthe same length
Cshorter
Dhalf as long
Show answer and explanation
Correct answer: (A) longer
Explanation: Step 1: Raising the confidence level raises the area (1 − α)/2 from 0.475 to 0.495, so the critical value Zα/2 becomes larger than 1.96.
Step 2: E = (σ/√n) × Zα/2 increases, so the length 2E increases.
∴ the second interval is longer.
Question 8Hard
In a study of pulse rate, a sample of 49 females has arithmetic mean 76.5, and the standard deviation of the female population is 12.5; a sample of 64 males has arithmetic mean 72, and the standard deviation of the male population is 12. The 95% confidence interval (critical value 1.96) is calculated for each population. The intersection of the two intervals is ……
A]69.06, 80[
B]72, 76.5[
Cthe two intervals have nothing in common
D]73, 74.94[
Show answer and explanation
Correct answer: (D) ]73, 74.94[
Explanation: Step 1: Females: E = 1.96 × 12.5/√49 = 3.5, so the interval is ]73, 80[.
Step 2: Males: E = 1.96 × 12/√64 = 2.94, so the interval is ]69.06, 74.94[.
∴ the intersection runs from the larger lower limit 73 to the smaller upper limit 74.94: ]73, 74.94[ (]69.06, 80[ is a trap: it is the union of the two intervals).
Question 9Hard
From a random sample taken from a normal population with standard deviation 12, two confidence intervals for the population mean are calculated: at the 95% level (critical value 1.96) and at the 99% level (critical value 2.58). The length of the second interval is greater than the length of the first by 1.86. Then the sample size = ……
A16
B64
C8
D256
Show answer and explanation
Correct answer: (B) 64
Explanation: Step 1: Length of an interval = 2 × Zα/2 × σ/√n, so the difference = 2 × (2.58 − 1.96) × 12/√n = 1.86.
Step 2: 1.24 × 12/√n = 1.86, so √n = 14.88 ÷ 1.86 = 8.
∴ n = 8² = 64 (8 is a trap: it is √n; 16 takes 1.86 as the difference between the two errors instead of the two lengths).
Question 10Hard
A factory wants to estimate the mean weight of its packets with a 95% confidence interval (critical value Zα/2 = 1.96) whose length is 1.96 grams. If the standard deviation of the packet weights is 5 grams, then the required sample size = ……
A100
B25
C10
D400
Show answer and explanation
Correct answer: (A) 100
Explanation: Step 1: Length of the interval = 2E, so E = 1.96 ÷ 2 = 0.98.
Step 2: 0.98 = (5/√n) × 1.96, so √n = 9.8 ÷ 0.98 = 10.
∴ n = 10² = 100 (25 is a trap: it takes the length 1.96 as E; 10 is √n).