Inferential Statistics: Population, Sample, Parameter, t-Test

A school has 1,200 students. To estimate the average time its students spend on homework each day, the principal asks 80 students chosen at random. In this study, what is the population?

1. aThe 80 students who were asked
2. bAll 1,200 students of the school
3. cThe average homework time of the 80 students
4. dThe average homework time of all 1,200 students

In a factory of 2,000 workers, 50 workers are chosen at random and their mean daily wage is found to be ₹ 640. The value ₹ 640 is a

1. aparameter, because it describes the workers of the factory
2. bpopulation, because it covers the wages of the workers
3. cstatistic, because it is worked out from a sample
4. dhypothesis, because it is a claim about the mean wage

A list of 600 households is numbered 1 to 600. A sample of 40 households is to be drawn by systematic random sampling, and the random start chosen from the first interval is household 7. Which households are the first four in the sample?

1. a7, 22, 37 and 52
2. b7, 47, 87 and 127
3. c7, 14, 21 and 28
4. d15, 30, 45 and 60

A one-sample t-test is carried out on a random sample of 16 observations. What are the degrees of freedom?

1. a16
2. b17
3. c14
4. d15

A random sample of 16 bulbs has a mean life of 1,040 hours and a standard deviation \(s = 80\) hours (calculated with divisor \(n - 1\)). To test \(H_0: \mu = 1000\) hours, what is the value of the t statistic \(t = \dfrac{\bar{x} - \mu}{s/\sqrt{n}}\)?

1. a0.5
2. b8
3. c2
4. d0.125

A random sample of 16 observations has standard deviation \(s = 8\) (calculated with divisor \(n - 1\)). It is used to test \(H_0: \mu = 50\) against \(H_1: \mu \ne 50\) at the 5% level of significance; the tabulated value of \(t\) for 15 degrees of freedom is 2.131. \(H_0\) will be rejected if the sample mean is greater than a certain value (or the same distance below 50). What is that value, to two decimal places?

1. a51.07
2. b54.26
3. c67.05
4. d52.13

A company claims that its batteries last 300 hours on average. A consumer group tests whether the true mean life differs from 300 hours. Which pair of hypotheses should it use?

1. a\(H_0: \mu = 300\), \(H_1: \mu \ne 300\)
2. b\(H_0: \mu \ne 300\), \(H_1: \mu = 300\)
3. c\(H_0: \bar{x} = 300\), \(H_1: \bar{x} \ne 300\)
4. d\(H_0: \mu = 300\), \(H_1: \mu > 300\)

A population has mean 60 and standard deviation 12. By the Central Limit Theorem, the mean of random samples of size 36 is approximately normally distributed with

1. amean 60 and standard deviation 12
2. bmean 60 and standard deviation \(\dfrac{1}{3}\)
3. cmean 10 and standard deviation 2
4. dmean 60 and standard deviation 2

A random sample of 5 observations is 8, 10, 12, 9 and 11. It is used to test \(H_0: \mu = 12\). Using \(s = \sqrt{\dfrac{\sum (x - \bar{x})^{2}}{n - 1}}\) and \(t = \dfrac{\bar{x} - \mu}{s/\sqrt{n}}\), what is \(|t|\)?

1. a\(\sqrt{10}\)
2. b\(\dfrac{4\sqrt{10}}{5}\)
3. c\(2\sqrt{2}\)
4. d\(\dfrac{2\sqrt{10}}{5}\)

To estimate the average monthly income of families in a city, a surveyor interviews only families living in one expensive gated community. What is wrong with this sample?

1. aIt is biased: most families in the city had no chance of being chosen
2. bIt is too large: a sample should have fewer than 30 families
3. cNothing: any group of families from the city is a representative sample
4. dIt is a census, so it cannot be used to estimate an average

Assertion (A): The mean mark of 50 students chosen at random from a school is only an estimate of the mean mark of all the students of the school.

Reason (R): A statistic varies from one sample to another, while a parameter has one fixed value for the population.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): For a one-sample t-test on 12 observations, the number of degrees of freedom is 11.

Reason (R): The t-test is used for small samples when the population standard deviation is not known.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): In a t-test, the null hypothesis is rejected when the calculated value of \(|t|\) is less than the tabulated value.

Reason (R): The tabulated value of \(t\) depends on the degrees of freedom and the level of significance.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. I. A census collects data from every member of the population.
2. II. II. In simple random sampling, every member of the population has an equal chance of being selected.
3. III. III. A sample becomes representative simply by being large, even if all its members come from one group.

Which of the statements above about sampling is/are correct?

1. aOnly I
2. bOnly I and III
3. cOnly I and II
4. dI, II and III

1. I. I. The Central Limit Theorem holds only when the population itself is normally distributed.
2. II. II. By the Central Limit Theorem, the distribution of the sample mean is close to normal when the sample size is large (usually 30 or more).
3. III. III. The sampling distribution of the mean can be studied by drawing many samples of the same size and plotting their means.

Which of the statements above about the Central Limit Theorem and sampling distributions is/are correct?

1. aOnly II
2. bOnly I and III
3. cI, II and III
4. dOnly II and III

1. I. I. A one-sample t-test checks whether a sample mean differs significantly from an assumed value of the population mean.
2. II. II. In a t-test, it is the alternative hypothesis, not the null hypothesis, that is assumed true and then tested.

Which of the statements above about the t-test is/are correct?

1. aOnly II
2. bOnly I
3. cBoth I and II
4. dNeither I nor II

Match each term in List I with its meaning in List II:

1. a(i)-Q, (ii)-P, (iii)-S, (iv)-R
2. b(i)-R, (ii)-S, (iii)-P, (iv)-Q
3. c(i)-Q, (ii)-S, (iii)-P, (iv)-R
4. d(i)-S, (ii)-Q, (iii)-P, (iv)-R

Each row of List I gives a sample mean \(\bar{x}\), a sample standard deviation \(s\) and a sample size \(n\) for testing \(H_0: \mu = 50\). Match each with \(|t|\) in List II, where \(t = \dfrac{\bar{x} - \mu}{s/\sqrt{n}}\):

1. a(i)-R, (ii)-S, (iii)-P, (iv)-Q
2. b(i)-R, (ii)-P, (iii)-S, (iv)-Q
3. c(i)-S, (ii)-R, (iii)-P, (iv)-Q
4. d(i)-Q, (ii)-S, (iii)-P, (iv)-R

A news channel wants to predict the result of an election in a city. Which plan gives a representative sample of the city's voters?

1. aInterviewing 500 people who work in the channel's own office
2. bInterviewing 500 voters picked using random numbers from the city's full voters' list
3. cUsing the first 500 replies to a poll on the channel's website
4. dInterviewing 500 voters from the ward where one candidate lives

A juice shop says each glass holds 250 ml. Riya measures 9 glasses chosen at random and finds a mean of 245 ml with \(s = 6\) ml (divisor \(n - 1\)). She tests \(H_0: \mu = 250\) against \(H_1: \mu \ne 250\) at the 5% level; the tabulated value of \(t\) for 8 degrees of freedom is 2.306. What should she conclude?

1. a\(|t| = 2.5\); do not reject \(H_0\): the mean does not differ significantly from 250 ml
2. b\(|t| \approx 0.83\); do not reject \(H_0\): the mean does not differ significantly from 250 ml
3. c\(|t| = 7.5\); reject \(H_0\): the mean differs significantly from 250 ml
4. d\(|t| = 2.5\); reject \(H_0\): the mean differs significantly from 250 ml