### Calculate the correlation coefficient and test

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Statistics Revision

1. A chemical compound is packed in 5kg packets. The packer has set his machinery to give bags of average weight 5.1 kg and he knows that the distribution of packet weights will be normal with standard deviation 0.05 kg. Calculate

(i) the probability that a particular packet weighs less than 5kg? (ii) the percentage of packets weighing more than 5.15kg?

2. A researcher was investigating the size of particulate emitted from a particular industrial process. When examined with a microscope it was found that the mean and standard deviation of the maximum diameter of the observed particulate was 7.6 and 0.9 microns respectively. Assuming that the maximum diameters are normally distributed about this mean what proportion of particles would have maximum diameters

(i) Greater than 9.4 microns?
(ii) Less than 7.6 microns?
(iii) Between 7.7 microns and 7.9 microns.

3. Sixty-four observations are selected at random from a normal distribution with mean 10 and standard deviation 5 and their mean is calculated. What is the probability that this mean is larger than 10.5?

4. Two hundred observations are selected at random from a distribution whose mean is 5 and variance 8. Find the probability of obtaining a sample mean less than 4.99.

5. In an investigation, six replicate determinations for aprinocid were made on a feed premix. The results were:
10.4, 10.4, 10.6, 10.3, 10.5,10.5

Calculate the mean and standard deviation of the sample. Assuming that the standard deviation, σ, is known to be 0.1, calculate the 95% and 99% confidence intervals for the mean.

6. Ten measurements of the ratio of two peak areas in a liquid chromatography experiment gave the following values:

0.2911, 0.2898, 0.2923, 0.3019, 0.2997, 0.2961, 0.2947, 0.2986, 0.2902, 0.2882

Calculate the mean, standard deviation and 95% and 99% confidence intervals.

7. Seven measurements of the pH of a buffer solution (x10-2) gave the following results:

512, 520, 515, 517, 516, 519, 515

Calculate the 95% and 99% confidence limits for the true ph.

Seminar

1. The specified level for a particular trace element within a processed food substance is l0ppm. A sample was taken at random from each of 10 batches and the results (ppm) were:

9.4, 8.9, 7.6, 7.8, 8.2, 8.0, 7.9, 8.3, 7.3, 8.5

Given that the data comes from a population with standard deviation 0.5, use a 5% significance test to check whether the level of this element is 10ppm.

Given no prior knowledge about the population use a 5% significance test to check whether the level of this element is 10ppm.

2. In order to evaluate a spectrophotometric method for the determination of titanium, the method was applied to alloy samples containing different certified amounts of titanium. The results (%Ti) are shown below:

 Sample Certified Value Mean from Sample Standard deviation from sample 1 0.496 0.482 0.0257 2 0.995 1.009 0.0248

For each alloy 8 replicate determinations were made. For each alloy, test whether the mean value differs significantly from thecertified value.

Single sample (one-sided test)

3. Two nematode species, G. primumand G. secundum, are described as having ova which are morphologically very similar but with different size distributions. Ova of species Primumare said to have an average length of 50 microns while ova of species Secundumhave an average length of less than 50 microns. Twelve ova of this morphological type are found in the faeces of a patient. The length of the ova are in microns:

37.2, 38.6, 41.2, 42.4, 44.8, 46.3, 48.1, 49.4, 49.7, 50.4, 51.6, 52.7

Carry out a hypothesis test to determine whether the Secundum species is affecting the patient.

Two samples t-tests

4. The table below shows results concerning the extraction and determination of tin in food stuffs (Analyst 1983 108 109). The results give the levels of tin recovered from the same product after boiling for different times in an open vessel.

Boiling Time (Mins) 30?75

Tin Found (mg/kg)?57, 57, 55, 56, 56, 55, 56, 55 51, 60, 48, 32, 46, 58, 56, 51

i) Use the F-test to check whether the variability (variance) is the same for both boiling times at the 5% significance level.

ii) ii) If you find that the variability is different you are unable to use the two sample t-test and must look up a more specialist statistical test. (Minitab can carry out the alternative test).

If you find that the variability is the same carry out a two sample t-test to check whether the mean levels of tin found are the same at the 5% significance level. iii) Comment on your results.

5. The weight gains of uninfected rats are to be compared with the weight gains of rats infected with a coccidan parasite. Six week weight gains (in grams) for the rats are shown below:

Uninfected 136 146 104 119 124 161 107 83 113 129 97 123

Infected 70 118 101 85 107 132 94

It is proposed to use a two sample t-test.

i) Do these data indicate that the assumption of equality of variances is likely to be valid? Check whether the two variances are equal using an F-test at the 5% significance level.

ii) ii) Carry out two Sample t-test. State your conclusions.

6. The diagnosis of Schsilosoma haemtlobiuminfections is usually based on finding the characteristic ova in the urine of the patient. Counts of the number of ova per urine sample give an indication of the number of worms present and hence the severity of the infection. It is thought that the sensitivity of this test varies with the time of collection of the urine sample, the following data show the number of ova (in thousands) passed in the urine at different times of the day by the same patient:

a.m. 6.9 2.7 6.2 2.6 2.2 1.3 5.5 2.2 1.2

p.m. 23.3 10.8 16.9 20.0 18.2 20.5 11.8 6.8 13.2

Such variation in egg output makes the estimation of the number of worms rather difficult. Do these data present evidence that the variance (standard deviation) of the egg output is less at one time than another? Use an F-test at the 5% significance level and report your conclusions.

7. In a plant manufacturing a nitrogenous fertiliser a limitation to increased output was the rate of filtration, on rotary filters, to separate the fertiliser solution from insoluble by-products. Laboratory experiments were carried out in an attempt to make the magma more easily filtered and a standard procedure was established for estimating the ease of filtration, consisting simply of filtering a given volume through a standard filter paper under a standard suction and observing the time for filtration. It was necessary to confirm that this test gave a reasonable indication of the ease of filtration an the plant filters. Samples of magma were taken at a time when filtering was judged to be 'fair' and also at a time when it was judged to be 'fairly good', this being part of a more extensive investigation covering other conditions also. The results are given below:

‘Fair' Period 8.4 9.8 12.2 12.6 13.0 9.2 13.6
‘Fairly Good' period 9.0 10.2 9.6 4.4 7.0

i) Use an F-test to check whether the variability of the two periods is the same at the 5% significance level.

ii) If the F-test is not significant at the 5% significance level, carry out a two-sided t-test to check whether the means are the same. Again use the 5% significance level.

Seminar

1. Samples of two species of grass were taken from a small area and analysed for fluoride content (ppm). The results are shown below:

 Grass A 15 13 16 12 Grass B 10 11 12 14 9 8

Test whether the median fluoride is the same for both grasses.

2. A manufacturer uses a large amount of a certain chemical. Since there are just two suppliers of this chemical, the manufacturer wishes to test whether the percentage of two contaminants is the same for the two sources against the alternative that there is a difference in the percentage of contaminants for the two suppliers. Data from two independent random samples are given below:

 Supplier A 0.86 0.69 0.72 0.65 1.13 0.65 1.18 0.45 1.41 0.5 1.04 0.41 Supplier B 0.55 0.4 0.22 0.58 0.16 0.07 0.09 0.16 0.26 0.36 0.2 0.15

State the hypotheses used. Use the appropriate test at the 5% significance level.
Need to know:

i) How to use the stem and leaf diagram to comment on the shape of the distribution

ii) When to use the Mann - Whitny U test

Seminar

(a) A pilot survey is carried out to ascertain whether there is an association between father's occupation and school leaving age and the following results are obtained.

Leaving Age (years) 16 17

Father's occupation                           16               17             18 and over

Professional                                          5                20              25

Non-professional                                25               80             45

Is there evidence of an association between school leaving age and father's occupation? Test at the 5% level.

(a) Sulphate content of boiler water may have some effect on the cracking of boilerplates. Test the following data that was collected to examine this possibility. Test at the 5% level.

 SO4(ppm) 0 200 400 600 Uncracked 37 43 26 44 Cracked 12 30 19 15

Seminar 7

1. Four groups of mice were given five daily injections of oestrone and each mouse was measured for the increase (mm) of the inter-hip gap 24hours after the injection. The results are given below:

 Group Increase(mm) 1 0.15 0.5 0.4 0.4 0.3 2 1.9 2.3 1.35 1.5 1.4 3 2 2.2 1.2 1.4 2.2 4 1.5 2.5 2.5 1.5 1.7

a Analyze the data to investigate whether there are differences between the four groups.

c Use Minitab to check your results.

2. The table below shows the number of days to death of 31 mice inoculated with three strains of typhoid organism.

 9D 11C DSC1 2 8 8 4 7 6 6 5 12 5 8 6 2 4 8 4 7 7 4 6 10 3 10 5 5 9 7 5 11 9 3

a Explain why this is an example of a completely randomised design.

b Analyze the data by hand to investigate whether the time to death depends on the strain of typhoid.

c Use Minitab to check your results.

3. A machine is used to produce a headache tablet for a pharmaceutical firm. Samples are taken from the machine periodically to check for quality. The following table gives the amount of a particular chemical found in each sample of tablets.

 Time Amount of Chemical 8 5.6 5.7 5.5 5.6 5.8 5.9 5.4 12 5.1 5.2 5.5 5.4 5.4 16 5.5 5.3 5.8 5.6 5.7 5.5 20 5.2 5.2 5.3 5.3 5.3 5.1 5.4

Use Minitab to test whether there is a difference in the amount of the chemical in the tablets as the day progresses.

Seminar

(a) In an experiment to compare the percentage efficiency of different chelating agents in extracting metal ions from aqueous solution, a randomized block design was used. The results are given below:

 Chelating Agent Day A B C D 1 84 80 83 79 2 79 77 80 79 3 83 78 80 78

Analyze the data by hand to investigate whether there are any differences between the chelating agents and also whether the day the experiment is performed affects results. Analyse the results at the 5% significance level

2. Three different methods of analysis (M1, M2 and M3) were used to determine the amount of a certain constituent in a sample (ppm). Each method is performed once by each of 5 analysts and the results are given below:

 Analyst Method 1 2 3 4 5 M1 7 6.9 6.8 7.1 6.9 M2 6.5 6.7 6.5 6.7 6.6 M3 6.6 6.2 6.4 6.3 6.4

a. Explain why a randomized block design was used.

b. Analyze the data by hand and test at the 5% significance level for differences between the methods and the analysts.

Seminar

1. The following data show the nitrogen levels (in ppm) in water before and after treatment in a chemical process:

 Before (X) After (Y) Before (X) After (Y) 18.2 49.2 25 23 21.9 30 13 17 5.2 16 76 79 20.4 47.8 59 66 2.8 3.4 25.6 46.8 21 16.8 50.4 84.9 11.3 10.7 41.2 65.2 6.1 5.7 21 52

a. Input these data into MINITAB.
b. Draw a scatterplot.
c. Is a linear relationship between these levels visible?
d. By hand, calculate the correlation coefficient and test whether it is zero.
e. By hand, calculate the intercept and gradient of the least squares regression ?line.
f. Use Minitab to find the correlation coefficient and to fit a regression line of ?before levels on after levels.
g. Comment on the percentage goodness of fit achieved.
h. Carry out a residual analysis. What do you conclude?
i. If the before level is 25.0ppm. Use your results to predict the after nitrogen ?level.

2. Sixteen observations on the viscosity of a polymer (y) and a process variable, reaction temperature (x) were made and are shown in the table below:

 Viscosity 2256 2340 2426 2293 2330 2368 2250 2409 Temperature 80 93 100 82 90 99 81 96 Viscosity 2364 2379 2440 2364 2404 2317 2309 2328 Temperature 94 93 97 95 100 85 86 87

Input these data into MINITAB.

(b) Plot viscosity against reaction temperature. What type of relationship do you ?observe?

(c) Calculate the correlation coefficient and test whether it is zero. Does this ?confirm your previous conclusions?

(d) Fit a linear regression. State your regression line and the percentage ?goodness of fit.

(e) For a temperature of 90 C use your regression line to predict the viscosity.

(f) Carry out a residual analysis. What do you conclude?

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