VCE General Maths Data Analysis 2023 Mini Test 1

Question 1 (9 marks) [2023 Exam 2 Q1]

Data was collected to investigate the use of electronic images to automate the sizing of oysters for sale.

The variables in this study were:

• • ID: identity number of the oyster
• • weight: weight of the oyster in grams (g)
• • volume: volume of the oyster in cubic centimetres (cm³)
• • image size: oyster size determined from its electronic image (in megapixels)
• • size: oyster size when offered for sale: small, medium or large

The data collected for a sample of 15 oysters is displayed in Table 1.

Table 1

ID	Weight (g)	Volume (cm³)	Image size (megapixels)	Size
1	12.9	13.0	5.1	large
2	11.4	11.7	4.8	medium
3	17.4	17.4	6.5	large
4	6.8	7.2	2.9	small
5	9.6	10.1	3.7	medium
6	15.5	15.6	5.7	large
7	9.7	9.9	4.0	small
8	7.0	7.5	2.7	small
9	12.6	12.7	5.5	medium
10	12.5	12.7	5.0	medium
11	10.1	10.5	3.9	medium
12	10.6	10.8	4.1	medium
13	13.0	13.1	5.3	large
14	8.1	8.5	3.5	small
15	14.1	14.2	5.3	large

a. Write down the number of categorical variables in Table 1. 1 mark

b. Determine, in grams:

i. the mean weight of all the oysters in this sample 1 mark

mean =

ii. the median weight of the large oysters in this sample. 1 mark

median =

c. When a least squares line is used to model the association between oyster weight and volume, the equation is:

\(\textit{volume} = 0.780 + 0.953 \times \textit{weight}\)

i. Name the response variable in this equation. 1 mark

ii. Complete the following sentence by filling in the box provided. 1 mark

This equation predicts that, on average, each 10 g increase in the weight of an oyster is associated with a cm³ increase in its volume.

d. A least squares line can also be used to model the association between an oyster’s volume, in cm³, and its electronic image size, in megapixels. In this model, image size is the explanatory variable.

Using data from Table 1, determine the equation of this least squares line. Use the template below to write your answer. Round the values of the intercept and slope to four significant figures. 2 marks

\(=\) \(+\) \(\times\)

e. The number of megapixels needed to construct an accurate electronic image of an oyster is approximately normally distributed.

Measurements made on recently harvested oysters showed that:

• • 97.5% of the electronic images contain less than 4.6 megapixels
• • 84% of the electronic images contain more than 4.3 megapixels.

Use the 68–95–99.7% rule to determine, in megapixels, the mean and standard deviation of this normal distribution. 2 marks

mean =

standard deviation =