Solved: A psychologist records the number of digits of π that a sample of IB Mathematics... [IB Mathematics Applications & Interpretation (Math AI)]

A psychologist records the number of digits of π that a sample of IB Mathematics higher level candidates could recall.

Question

HLPaper 1

A psychologist records the number of digits of π that a sample of IB Mathematics higher level candidates could recall.

d	2	3	4	5	6	7
Frequency	2	6	24	21	11	3

Find an unbiased estimate of the population mean of d.

[1]

Verified

Solution

Let $\bar{x}$ be the sample mean $\bar{x} = \frac{2(2) + 3(6) + 4(24) + 5(21) + 6(11) + 7(3)}{67}$ $\bar{x} = \frac{134}{67} = 4.687...$

The sample mean is an unbiased estimate of the population mean.

Find an unbiased estimate of the population variance of d.

[2]

Verified

Solution

The unbiased estimate of population variance is: $s^2 = \frac{\sum(x-\bar{x})^2}{n-1}$ where n = 67

Calculating $\sum(x-\bar{x})^2$ : $(2-4.687)^2(2) + (3-4.687)^2(6) + (4-4.687)^2(24) + (5-4.687)^2(21) + (6-4.687)^2(11) + (7-4.687)^2(3)$

$s^2 = 1.243...$

The psychologist has read that in the general population people can remember an average of 4.4 digits of π. The psychologist wants to perform a statistical test to see if IB Mathematics higher level candidates can remember more digits than the general population. H₀: μ = 4.4 is the null hypothesis for this test. State the alternative hypothesis.

[1]

Verified

Solution

H₁: μ > 4.4

Note: Must use correct notation (μ) and inequality sign (>)

Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion. Use a 5% significance level.

[4]

Verified

Solution

This is a one-tailed z-test (or t-test) as n > 30

Test statistic: $z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

where $\bar{x} = 4.687$ , $\mu_0 = 4.4$ , $s = \sqrt{1.243} = 1.115$ , $n = 67$

$z = \frac{4.687 - 4.4}{\frac{1.115}{\sqrt{67}}} = 2.127$

Critical value at 5% significance level (one-tailed) is 1.645

Since 2.127 > 1.645, we reject H₀

Conclusion: There is sufficient evidence at the 5% significance level to conclude that IB Mathematics HL candidates can remember more digits of π than the general population.

A psychologist records the number of digits of π that a sample of IB Mathematics higher level candidates could recall.

Question

HLPaper 1

A psychologist records the number of digits of π that a sample of IB Mathematics higher level candidates could recall.

d	2	3	4	5	6	7
Frequency	2	6	24	21	11	3

Find an unbiased estimate of the population mean of d.

[1]

Verified

Solution

Let $\bar{x}$ be the sample mean $\bar{x} = \frac{2(2) + 3(6) + 4(24) + 5(21) + 6(11) + 7(3)}{67}$ $\bar{x} = \frac{134}{67} = 4.687...$

The sample mean is an unbiased estimate of the population mean.

Find an unbiased estimate of the population variance of d.

[2]

Verified

Solution

The unbiased estimate of population variance is: $s^2 = \frac{\sum(x-\bar{x})^2}{n-1}$ where n = 67

Calculating $\sum(x-\bar{x})^2$ : $(2-4.687)^2(2) + (3-4.687)^2(6) + (4-4.687)^2(24) + (5-4.687)^2(21) + (6-4.687)^2(11) + (7-4.687)^2(3)$

$s^2 = 1.243...$

[1]

Verified

Solution

H₁: μ > 4.4

Note: Must use correct notation (μ) and inequality sign (>)

Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion. Use a 5% significance level.

[4]

Verified

Solution

This is a one-tailed z-test (or t-test) as n > 30

Test statistic: $z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

where $\bar{x} = 4.687$ , $\mu_0 = 4.4$ , $s = \sqrt{1.243} = 1.115$ , $n = 67$

$z = \frac{4.687 - 4.4}{\frac{1.115}{\sqrt{67}}} = 2.127$

Critical value at 5% significance level (one-tailed) is 1.645

Since 2.127 > 1.645, we reject H₀

Conclusion: There is sufficient evidence at the 5% significance level to conclude that IB Mathematics HL candidates can remember more digits of π than the general population.

A psychologist records the number of digits of π that a sample of IB Mathematics higher level candidates could recall.

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A psychologist records the number of digits of π that a sample of IB Mathematics higher level candidates could recall.

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