Sampling, Inference and the t-Test for CUET Applied Maths

AnswerB · All 20 lakh voters of the city

1. The population is the whole group the question is about; the sample is the part actually studied.

2. Here the population is all 20 lakh voters, and the 500 asked are the sample.

AnswerA · $20$

1. The interval is $k = \frac{N}{n}$, the population size over the sample size.

2. $k = \frac{600}{30} = 20$: every 20th student after a random start.

AnswerA · Only I

1. I is the definition of simple random sampling. True.

2. II: a large sample drawn only from one group stays biased; size does not cure a bad method. False.

AnswerA · Only I

1. I: more randomly chosen units average out chance differences. True.

2. II: volunteers choose themselves and often differ from the rest, so such a sample can be biased. False.

AnswerC · 1 to 25

1. The start must fall in the first interval, so that every unit has the same chance.

2. So it is drawn at random from units 1 to 25.

AnswerD · The 50 bulbs tested

1. The population is every unit we want to know about: all 5,000 bulbs.

2. The sample is the part actually studied: the 50 bulbs tested.

AnswerA · $30$

1. Interval $k = \frac{N}{n}$.

2. $k = \frac{1{,}200}{40} = 30$: every 30th household after a random start.

AnswerB · 20 names drawn by lottery from a list of all 300 students

1. In a simple random sample every unit has the same chance of being chosen.

2. A lottery over the full list gives every student that equal chance; the others favour early risers, players or toppers.

AnswerD · All 1,200 students of the school

1. The population is the whole group the question is about.

2. She asks about her school's 1,200 students.

AnswerC · The mean daily screen time of all 1,200 students

1. A parameter is a value describing the whole population.

2. The school-wide mean is that value; she does not know it yet.

AnswerD · A random start among the first 10 names, then every 10th name

1. Systematic sampling picks one unit at random from the first $k$, then every $k$th unit after it.

2. Drawing from a mixed box is simple random sampling; the last two are not random at all.

AnswerC · $67$

1. The chosen units are 7, $7 + 20$, $7 + 40$, $7 + 60$, …

2. The 4th is $7 + 3 \times 20 = 67$.

AnswerB · Members of a gym, to estimate how much the adults of a city exercise

1. A sample is representative when every part of the population has a fair chance of being in it.

2. Gym members exercise more than most adults, so they would overstate the city's exercise: a biased sample.

AnswerC · $64$

1. The chosen units are 4, 19, 34, 49, 64, …: the $k$th is $4 + (k - 1) \times 15$.

2. The 5th is $4 + 4 \times 15 = 64$.

AnswerD · Students found in the city library on one afternoon

1. A biased sample systematically over- or under-represents part of the population.

2. Students in a library are likely to read more than average, so that sample overstates reading time.

AnswerB · As an estimate of the school's mean screen time

1. A statistic from a random sample estimates the matching parameter.

2. Another sample would give a slightly different mean, so 3.2 hours is an estimate, not the exact value.

AnswerD · Taking 60 members of the school's computer club

1. A sample is biased when some students are more likely to be chosen in a way linked to what is measured.

2. Computer-club members likely spend more time on screens, so they would push the estimate up.

AnswerD · A parameter

1. A parameter describes a whole population; a statistic describes a sample.

2. Every student is included, so the average is a parameter.

AnswerC · A-II, B-IV, C-I, D-III

1. Greek letters stand for parameters: $\mu$ for the population mean, $\sigma$ for its standard deviation.

2. Roman letters stand for statistics: $\bar{x}$ for a sample mean, $s$ for a sample standard deviation.

AnswerC · $2$

1. Standard error $= \frac{\sigma}{\sqrt{n}}$.

2. $\frac{20}{\sqrt{100}} = \frac{20}{10} = 2$.

AnswerB · The sampling distribution of the mean

1. Each possible sample gives its own mean, and these means form a distribution.

2. That is the sampling distribution of the mean, linking the population to a single sample.

AnswerB · A statistic

1. A statistic describes a sample; a parameter describes the whole population.

2. This mean comes from 200 sampled families, so it is a statistic.

AnswerA · The mean bill of ₹1,850 of the 900 sampled households

1. A statistic is worked out from the sample; a parameter describes the whole population.

2. The ₹1,850 comes from the 900 sampled households, so it is a statistic.

AnswerC · ₹10

1. Standard error $= \frac{\sigma}{\sqrt{n}} = \frac{300}{\sqrt{900}}$.

2. $\frac{300}{30} = 10$ rupees.

AnswerB · $2$

1. The standard error of the mean is $\frac{\sigma}{\sqrt{n}}$.

2. $\frac{12}{\sqrt{36}} = \frac{12}{6} = 2$.

AnswerD · It is nearly normal, with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$

1. For a large sample, the sample mean is close to normally distributed, whatever the shape of the population.

2. Its distribution is centred at $\mu$, with spread $\frac{\sigma}{\sqrt{n}}$, smaller than the population's.

AnswerA · Only I

1. I: there is one population mean, but each sample has its own mean. True.

2. II: a sample mean is only an estimate and usually misses the population mean by a little. False.

AnswerA · Both A and R are true, and R is the correct explanation of A.

1. Both are true: a sample is only part of the population.

2. Because a statistic varies from sample to sample, the one sample in hand may sit above or below the truth; that is why A holds. R explains A.

AnswerD · $36$

1. $\frac{30}{\sqrt{n}} = 5$, so $\sqrt{n} = 6$.

2. $n = 36$.

AnswerD · Under $H_0$, such a result has under a 5% chance

1. Significance compares the result with what chance alone would give if $H_0$ held.

2. At 5%, a result is significant when chance would produce one so extreme less than 5% of the time.

AnswerC · $2$

1. The standard error is $\frac{\sigma}{\sqrt{n}}$: four times the sample size divides it by $\sqrt{4} = 2$.

2. $\frac{4}{2} = 2$.

AnswerA · About normal, with mean 80 and standard error 3

1. For a large sample, $\bar{x}$ is roughly normal with the population's mean, 80.

2. Its spread is the standard error $\frac{21}{\sqrt{49}} = 3$.

AnswerA · Both A and R are true, and R is the correct explanation of A.

1. R is true: a statistic is computed from a sample.

2. Different samples hold different units, so $\bar{x}$ differs: A is true because of R.

AnswerB · ₹5

1. $\frac{300}{\sqrt{3{,}600}} = \frac{300}{60}$.

2. ₹5: four times the sample halves the standard error.

AnswerC · $4$

1. The standard error is $\frac{\sigma}{\sqrt{n}}$, so it falls with the square root of $n$.

2. To halve it, $\sqrt{n}$ must double, so $n$ must be multiplied by 4.

AnswerA · ₹19.60

1. The standard error is ₹10.

2. $1.96 \times 10 = 19.6$ rupees.

AnswerB · $H_0: \mu = 500$

1. The null hypothesis says there is no difference: the population mean is the claimed value.

2. $H_0: \mu = 500$.

AnswerC · Do not reject $H_0$

1. $H_0$ is rejected only when $\lvert t \rvert$ is larger than the critical value.

2. A smaller $\lvert t \rvert$ means the data are consistent with $H_0$: do not reject it.

AnswerB · $H_0: \mu = 7$

1. The null hypothesis states the claim, with no difference: the population mean equals 7.

2. So $H_0: \mu = 7$.

AnswerC · $2$

1. $t = \frac{74 - 70}{10/\sqrt{25}} = \frac{4}{2}$.

2. $t = 2$.

AnswerD · $H_1: \mu \ne 500$

1. A non-directional alternate hypothesis allows a difference either way.

2. $H_1: \mu \ne 500$.

AnswerC · $H_0: \bar{x} = 52$

1. A hypothesis is a claim about the population, so it is written with a parameter such as $\mu$.

2. $\bar{x} = 52$ is already known from the sample; there is nothing to test.

AnswerA · Only I

1. I is what the null hypothesis means. True.

2. II: a test can only reject the null hypothesis or fail to reject it; failing to reject is not proof. False.

AnswerB · The chance of rejecting $H_0$ when it is in fact true

1. The level of significance is the risk we accept of rejecting a true null hypothesis.

2. At 5%, a result as extreme as the critical value would happen by chance only 5 times in 100 if $H_0$ were true.

AnswerD · $H_1: \mu < 300$

1. The alternate hypothesis states what the researcher suspects instead of the claim.

2. 'Less than 300' gives the one-sided $H_1: \mu < 300$.

AnswerC · Both I and II

1. I: 'different' allows a departure in either direction, both tails. True.

2. II: a one-direction suspicion uses one tail. True.

AnswerB · I, II and IV only

1. I, II and IV describe how hypotheses are set up.

2. III is false: a test works with probabilities; rejecting $H_0$ at 5% still leaves a chance of error.

AnswerA · $H_1: \mu > 40$

1. The claim to be checked against is $H_0: \mu = 40$.

2. The hope is an increase, so $H_1: \mu > 40$, a one-sided test.

AnswerA · $H_0: \mu = 70,\ H_1: \mu > 70$

1. The null hypothesis is the 'no better than 70' position, $\mu = 70$.

2. The claim of more than 70 is the alternative, $H_1: \mu > 70$, a right-tailed test.

AnswerB · Reject $H_0$: $t = 2$ is above 1.711, so the data back the claim

1. A right-tailed test compares $t$ with the one-tailed critical value, 1.711.

2. $2 > 1.711$: reject $H_0$ and accept that the mean is above 70.

AnswerD · Do not reject $H_0$, since $t = 2$ is below 2.064

1. Two-tailed, the 5% is split between both tails, so the critical value rises to 2.064.

2. $t = 2 < 2.064$: do not reject $H_0$. The choice of tails can change the verdict.

AnswerC · $15$

1. For a one-sample t-test, the degrees of freedom are $n - 1$.

2. $16 - 1 = 15$.

AnswerC · $2$

1. Standard error $= \frac{s}{\sqrt{n}}$.

2. $\frac{14}{7} = 2$.

AnswerB · $2.5$

1. $t$ is the difference divided by the standard error.

2. $\frac{3}{1.2} = 2.5$.