IB Questionbank

Loading [MathJax]/jax/element/mml/optable/Latin1Supplement.js

Updates to Questionbank

User interface language: English | Español

Date	May 2019	Marks available	1	Reference code	19M.3.AHL.TZ0.Hsp_3
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Write down	Question number	Hsp_3	Adapted from	N/A

Question

In a large population of hens, the weight of a hen is normally distributed with mean $\mu$ kg and standard deviation $\sigma$ kg. A random sample of 100 hens is taken from the population.

The mean weight for the sample is denoted by $\bar X$ .

The sample values are summarized by $\sum {x = 199.8}$ and $\sum {{x^2} = 407.8}$ where $x$ kg is the weight of a hen.

It is found that $\sigma$ = 0.27 . It is decided to test, at the 1 % level of significance, the null hypothesis $\mu$ = 1.95 against the alternative hypothesis $\mu$ > 1.95.

State the distribution of $\bar X$ giving its mean and variance.

[1]

a.

Find an unbiased estimate for $\mu$ .

[1]

b.

Find an unbiased estimate for ${\sigma ^2}$ .

[2]

c.

Find a 90 % confidence interval for $\mu$ .

[3]

d.

Find the $p$ -value for the test.

[2]

e.i.

Write down the conclusion reached.

[1]

e.ii.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$\bar X \sim N\left( {\mu {\text{,}}\,\,\frac{{{\sigma ^2}}}{{100}}} \right)$ A1

Note: Accept $n$ in place of 100.

[1 mark]

a.

$\hat \mu = \frac{{\sum x }}{n} = \frac{{199.8}}{{100}} = 1.998$ A1

Note: Accept 2.00, 2.0 and 2.

[1 mark]

b.

${s_{n - 1}}^2 = \frac{n}{{n - 1}}\left( {\frac{{\sum {{x^2}} }}{n} - {{\bar x}^2}} \right) = \frac{{100}}{{99}}\left( {\frac{{407.8}}{{100}} - {{1.998}^2}} \right)$ (M1)

= 0.086864

unbiased estimate for ${\sigma ^2}$ is 0.0869 A1

Note: Accept any answer which rounds to 0.087.

[2 marks]

c.

90 % confidence interval is $1.998 \pm 1.660\sqrt {\frac{{0.0869}}{{100}}}$ (M1)

= (1.95, 2.05) A1A1

Note: FT their $\sigma$ from (c).

Note: Condone the use of the $z$ -value 1.645 since $n$ is large.

Note: Accept any values that round to 1.95 and 2.05.

[3 marks]

d.

$p$ -value is 0.0377 A2

Note: Award A1 for the 2-tail value 0.0754.

Note: Award A2 for 0.0377 and A1 for any other value that rounds to 0.038.

Note: FT their estimated mean from (b), note that 2 gives $p$ = 0.032(0).

[2 marks]

e.i.

accept the null hypothesis A1

Note: FT their $p$ -value.

[1 mark]

e.ii.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

[N/A]

e.i.

[N/A]

e.ii.

Syllabus sections

Topic 4—Statistics and probability » AHL 4.14—Linear transformation of a single RV, E(X) and VAR(X), unbiased estimators

Show 62 related questions

22M.2.AHL.TZ1.3e.i:
their mean standardized score.
22M.2.AHL.TZ1.3e.ii:
the standard deviation of their standardized score.
18M.3.AHL.TZ0.Hsp_1a:
Find the probability that a randomly chosen male bird weighs between 4.75 kg and 4.85 kg.
EXM.3.AHL.TZ0.8a.ii:
Find unbiased estimates for the population Variance.
EXM.3.AHL.TZ0.8a.i:
Find unbiased estimates for the population mean.
EXN.3.AHL.TZ0.1f:
Given this result, comment on the validity of the linear model used in part (a).
22M.1.AHL.TZ1.9b:
A statistician in the company suggests it would be fairer if the company passes the inspection when the mean weight of five randomly chosen bags is greater than $950 g$ .

Find the probability of passing the inspection if the statistician’s suggestion is followed.
22M.1.AHL.TZ2.9b:
Find an unbiased estimate of the population variance of $d$ .
21M.1.AHL.TZ1.14a:
Find the mean weight of a bag of apples.
20N.3.AHL.TZ0.Hsp_3a:
Assuming that the shopkeeper’s claim is correct, find the probability that the weight of six randomly chosen carrots is more than two times the weight of one randomly chosen broccoli.
22M.1.AHL.TZ2.9a:
Find an unbiased estimate of the population mean of $d$ .
EXM.3.AHL.TZ0.8f:
Perform a suitable test, at the 5% significance level, to determine if it is easier to achieve a distinction on the new exam. You should clearly state your hypotheses, the critical region and your conclusion.
18N.3.AHL.TZ0.Hsp_3b.i:
State the distribution of your test statistic, including the parameter.
EXM.3.AHL.TZ0.8d:
Use the normal distribution model to find the score required to pass.
18N.3.AHL.TZ0.Hsp_3c.iii:
State the p-value and interpret it in this context.
SPM.1.AHL.TZ0.9a:
Find ${s_{n - 1}}$ for this sample.
EXM.3.AHL.TZ0.8c:
Perform a suitable test, at the 5% significance level, to determine if the scores follow a normal distribution, with the mean and variance found in part (a). You should clearly state your hypotheses, the degrees of freedom, the p-value and your conclusion.
EXM.1.AHL.TZ0.59b.ii:
Calculate the value of the ${\chi ^2}$ statistic and state your conclusion using a 10% level of significance.
EXM.1.AHL.TZ0.59b.i:
State suitable hypotheses.
18N.3.AHL.TZ0.Hsp_1a:
${\text{E}}\left( {2X + 7Y} \right)$ .
19M.3.AHL.TZ0.Hsp_3d:
Find a 90 % confidence interval for $\mu$ .
18N.3.AHL.TZ0.Hsp_3c.i:
State suitable hypotheses for the test.
SPM.1.AHL.TZ0.9c:
The bags are labelled as being 1.5 kg mass. Comment on this claim with reference to your answer in part (b).
19M.3.AHL.TZ0.Hsp_2b:
Find the probability that the time taken by an employee to deal with a queue of three customers is less than nine minutes.
18N.3.AHL.TZ0.Hsp_3d:
Use an appropriate regression line to estimate the weight of a fish with length 360 mm.
EXM.3.AHL.TZ0.8b:
Show that the expected frequency for 20 < $x$ ≤ 4 is 31.5 correct to 1 decimal place.
21M.1.AHL.TZ2.10d:
Suggest, with justification, a valid conclusion that Talha could make.
EXM.3.AHL.TZ0.8g.i:
Find the probability of making a Type I error.
19M.3.AHL.TZ0.Hsp_3c:
Find an unbiased estimate for ${\sigma ^2}$ .
EXN.2.AHL.TZ0.6b.i:
both have a mass greater than $3.0 kg$ .
18N.3.AHL.TZ0.Hsp_3a.i:
Find the probability that a fish from this lake will have a weight of more than 560 grams.
19M.3.AHL.TZ0.Hsp_2c:
At the start of the day, one employee, Amanda, has a queue of four customers. A second employee, Brian, has a queue of three customers. You may assume they work independently.

Find the probability that Amanda’s queue will be dealt with before Brian’s queue.
18N.3.AHL.TZ0.Hsp_3b.iii:
State the conclusion of the test, justifying your answer.
EXM.1.AHL.TZ0.59a:
Calculate unbiased estimates of the population mean and the population variance.
18M.3.AHL.TZ0.Hsp_1c:
Two randomly chosen male birds and three randomly chosen female birds are placed on a weighing machine that has a weight limit of 18 kg. Find the probability that the total weight of these five birds is greater than the weight limit.
18N.3.AHL.TZ0.Hsp_3c.ii:
Find the product-moment correlation coefficient $r$ .
19M.3.AHL.TZ0.Hsp_3a:
State the distribution of $\bar X$ giving its mean and variance.
18N.3.AHL.TZ0.Hsp_3a.ii:
The maximum weight a hand net can hold is 6 kg. Find the probability that a catch of 11 fish can be carried in the hand net.
18N.3.AHL.TZ0.Hsp_1c:
${\text{E}}\left( {{X^2} - {Y^2}} \right)$ .
18N.3.AHL.TZ0.Hsp_3b.ii:
Find the p-value for the test.
19M.3.AHL.TZ0.Hsp_2a:
Find the probability that the time taken for a randomly chosen customer to be dealt with by an employee is greater than 180 seconds.
18M.3.AHL.TZ0.Hsp_1b:
Find the probability that the weight of a randomly chosen male bird is more than twice the weight of a randomly chosen female bird.
20N.3.AHL.TZ0.Hsp_3b:
Find the least value of $n$ required to ensure that the width of the confidence interval is less than $2 grams$ .
18N.3.AHL.TZ0.Hsp_1b:
Var $\left( {4X - 3Y} \right)$ .
19M.3.AHL.TZ0.Hsp_3e.i:
Find the $p$ -value for the test.
21M.1.AHL.TZ2.10c:
Find a $95 %$ confidence interval for $μ$ . You may assume that all conditions for a confidence interval have been met.
19M.3.AHL.TZ0.Hsp_3b:
Find an unbiased estimate for $\mu$ .
SPM.1.AHL.TZ0.9b:
Find a 95 % confidence interval for the population mean, giving your answer to 4 significant figures.
EXM.3.AHL.TZ0.8e:
Perform a suitable test, at the 5% significance level, to determine if there is a difference between the mean scores of males and females. You should clearly state your hypotheses, the p-value and your conclusion.
EXM.3.AHL.TZ0.8g.ii:
Given that $p = 0.2$ find the probability of making a Type II error.
20N.3.AHL.TZ0.Hsp_3d:
Given that the weights of the broccoli actually follow a normal distribution with mean $392 grams$ and variance $80 {grams}^{2}$ , find the probability of Anjali making a Type II error.
20N.3.AHL.TZ0.Hsp_3c:
Find the significance level for this test.
21M.1.AHL.TZ1.14c:
Find the probability that a bag selected at random weighs more than $1 kg$ .
21M.1.AHL.TZ2.10a:
Find an unbiased estimate for the mean number $(μ)$ of chocolates per packet.
21M.1.AHL.TZ1.14b:
Find the standard deviation of the weights of these bags of apples.
21M.1.AHL.TZ2.10b:
Use the formula $s_{n - 1}^{2} = \frac{Σ x^{2} - \frac{{(Σ x)}^{2}}{n}}{n - 1}$ to determine an unbiased estimate for the variance of the number of chocolates per packet.
EXN.3.AHL.TZ0.1e.ii:
Hence find upper and lower bounds for the number of fish in the lake when $t = 8$ .
EXN.2.AHL.TZ0.3a:
Use the given value of $s_{n}$ to find the value of $s_{n - 1}$ .
EXN.2.AHL.TZ0.6b.ii:
have a total mass greater than $6.0 kg$ .
EXN.3.AHL.TZ0.1d.i:
Show that an estimate for $Var (X)$ is $38.25$ .
EXN.3.AHL.TZ0.1d.ii:
Hence show that the variance of the proportion of marked fish in the sample, $Var (\frac{X}{300})$ , is $0.000425$ .
EXN.3.AHL.TZ0.1e.i:
Taking the value for the variance given in (d) (ii) as a good approximation for the true variance, find the upper and lower bounds for the proportion of marked fish in the lake.

Hide related questions

Topic 4—Statistics and probability » AHL 4.15—Central limit theorem

Topic 4—Statistics and probability » AHL 4.16—Confidence intervals

Topic 4—Statistics and probability » AHL 4.18—T and Z test, type I and II errors

Topic 4—Statistics and probability