Edexcel FS2 2020 June — Question 1 6 marks

Exam BoardEdexcel
ModuleFS2 (Further Statistics 2)
Year2020
SessionJune
Marks6
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Mark schemeDownload PDF ↗
TopicF-test and chi-squared for variance
TypeF-test two variances hypothesis
DifficultyStandard +0.3 This is a standard F-test for variance comparison with clearly stated hypotheses, given summary statistics, and straightforward application of the test procedure. While it's Further Maths content (FS2), the question is routine: calculate sample variances from given Sxx values, form the F-statistic, compare to critical value, and conclude. No conceptual challenges or novel problem-solving required—just methodical execution of a textbook procedure.
Spec2.02g Calculate mean and standard deviation5.05c Hypothesis test: normal distribution for population mean
1 Gina receives a large number of packages from two companies, \(A\) and \(B\). She believes that the variance of the weights of packages from company \(A\) is greater than the variance of the weights of packages from company \(B\). Gina takes a random sample of 7 packages from company \(A\) and an independent random sample of 10 packages from company \(B\). Her results are summarised below $$\bar { a } = 300 \quad \mathrm {~S} _ { a a } = 145496 \quad \bar { b } = 233.4 \quad \mathrm {~S} _ { b b } = 56364.4$$ [You may assume that the weights of packages from the two companies are normally distributed.]
Test Gina's belief. Use a \(5 \%\) level of significance and state your hypotheses clearly.
Question 1:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(H_0: \sigma_a^2 = \sigma_b^2 \quad H_1: \sigma_a^2 > \sigma_b^2\)B1 For both hypotheses in terms of \(\sigma\)
\(s_a^2 = \frac{145496}{6} = [24249.33...]\) and \(s_b^2 = \frac{56364.4}{9} = [6262.711...]\)M1 For at least one \(s^2\) calculation correctly attempted; \(s_a = 155.72...\) accept awrt 156, \(s_b = 79.137...\) accept awrt 79.1
\(F_{6,9} = \frac{"24249.33..."}{``6262.711..."} = 3.872...\)M1 For correct calculation of test statistic (ft their \(s^2\))
\(= 3.872...\)A1 For awrt 3.87
\(F_{6,9}(5\%\) one-tail) c.v. \(= 3.37\)B1 For correct cv awrt 3.37
Significant, there is evidence to support Gina's beliefA1 For correct conclusion mentioning Gina's belief (o.e.) 
## Question 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \sigma_a^2 = \sigma_b^2 \quad H_1: \sigma_a^2 > \sigma_b^2$ | B1 | For both hypotheses in terms of $\sigma$ |
| $s_a^2 = \frac{145496}{6} = [24249.33...]$ and $s_b^2 = \frac{56364.4}{9} = [6262.711...]$ | M1 | For at least one $s^2$ calculation correctly attempted; $s_a = 155.72...$ accept awrt 156, $s_b = 79.137...$ accept awrt 79.1 |
| $F_{6,9} = \frac{"24249.33..."}{``6262.711..."} = 3.872...$ | M1 | For correct calculation of test statistic (ft their $s^2$) |
| $= 3.872...$ | A1 | For awrt 3.87 |
| $F_{6,9}(5\%$ one-tail) c.v. $= 3.37$ | B1 | For correct cv awrt 3.37 |
| Significant, there is evidence to support Gina's belief | A1 | For correct conclusion mentioning Gina's belief (o.e.) |

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1 Gina receives a large number of packages from two companies, $A$ and $B$. She believes that the variance of the weights of packages from company $A$ is greater than the variance of the weights of packages from company $B$.

Gina takes a random sample of 7 packages from company $A$ and an independent random sample of 10 packages from company $B$. Her results are summarised below

$$\bar { a } = 300 \quad \mathrm {~S} _ { a a } = 145496 \quad \bar { b } = 233.4 \quad \mathrm {~S} _ { b b } = 56364.4$$

[You may assume that the weights of packages from the two companies are normally distributed.]\\
Test Gina's belief. Use a $5 \%$ level of significance and state your hypotheses clearly.

\hfill \mbox{\textit{Edexcel FS2 2020 Q1 [6]}}
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