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2.5 Statistical Hypothesis Testing

2.5 Statistical Hypothesis Testing

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Question 25

Marine biologists are investigating the correlation between water salinity (x x\,x in parts per thousand, ppt) and the dissolved oxygen concentration (y y\,y in mg/L) in a coastal estuary. The least squares regression line of y y\,y on x x\,x is established as:

y=12.3−0.16x y = 12.3 - 0.16x y=12.3−0.16x

The following summary data from 15 samples are recorded:

∑y=112.5,∑y2=862.0,∑x2=13900,n=15 \sum y = 112.5, \quad \sum y^2 = 862.0, \quad \sum x^2 = 13900, \quad n = 15 ∑y=112.5,∑y2=862.0,∑x2=13900,n=15
a.

Show that Syy=18.25S_{yy} = 18.25Syy​=18.25.

[2]
b.

Calculate SxxS_{xx}Sxx​.

[3]
c.

Determine the product moment correlation coefficient (PMCC) between x x\,x and yyy.

[2]
d.

A biologist claims that at a salinity level of 10 ppt, the dissolved oxygen concentration will be greater than 10 mg/L.

Explain how the biologist might have reached this conclusion using the regression model.

[2]
e.

Another researcher defines the typical range of observed salinity values using the formula:

range=mean±2.5×standard deviation \text{range} = \text{mean} \pm 2.5 \times \text{standard deviation} range=mean±2.5×standard deviation

Using this formula and the given summary statistics, find the minimum and maximum salinity values for this range.

[3]
f.

With reference to the range calculated in part (e), comment on the reliability of the biologist’s claim in part (d).

[2]
g.

A student suggests using the regression equation y=12.3−0.16xy = 12.3 - 0.16xy=12.3−0.16x to predict the salinity (xxx) of a sample where the oxygen concentration is 6.5 mg/L. Evaluate the student’s suggestion.

[1]
Markscheme

2.5 Statistical Hypothesis Testing Questions

  1. A Level
  2. /Maths
  3. /2.5 Statistical Hypothesis Testing

316 exam-style questions on OCR A Level Maths 2.5 Statistical Hypothesis Testing, covering 2.5.1 The language of hypothesis testing, 2.5.2 Hypothesis test for a binomial proportion, 2.5.3 Inference and significance level, 2.5.4 Sample mean as a random variable (A-level only), 2.5.5 Hypothesis test for the mean of a normal distribution (A-level only), 2.5.6 Pearson's product-moment correlation coefficient, and 2.5.7 Hypothesis test using Pearson's coefficient (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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