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One Sample t-test

By Sally Long,2014-12-12 01:10
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One Sample t-test ______________________________________________________________________ A manufacturer of fertiliser claims that on average one bag of his fertiliser will be sufficient to cover 14 square metres of lawn. A firm buying the bags takes a random sample of 12 bags and measures the area covered by each. The data are: 13.2 13.8 13.4 12.4 ..

    One Sample t-test

    ______________________________________________________________________ A manufacturer of fertiliser claims that on average one bag of his fertiliser will be sufficient to cover 14 square metres of lawn. A firm buying the bags takes a random sample of 12 bags and measures the area covered by each. The data are:

    13.2 13.8 13.4 12.4 14.3 13.2

    13.6 13.9 13.2 14.5 12.6 12.6

Is the manufacturer's claim true?

    i.e. Test H : ; = 14 against H : ; 14 01

    1. Enter the 12 data values into C1 in a Minitab worksheet and name the column

    area.

    2. Select

    Stat > Basic Statistics > 1-sample t

    and complete the dialog box as shown. In order to carry out the test with the appropriate alternative hypothesis, select the Options box and set the Alternative to

    Not equal.

The resulting output is

One-Sample T: area

Test of mu = 14 vs mu not = 14

Variable N Mean StDev SE Mean

    area 12 13.392 0.665 0.192

Variable 95.0% CI T P

    area ( 12.969, 13.814) -3.17 0.009

    Note that:

     The p-value is 0.009 < 0.05. This indicates that, at the 5% significance level,

    the null hypothesis can be rejected. There is evidence that the manufacturer

    claim is not correct.

     The 95% confidence interval indicates that the true mean is between 13.0 and

    13.8 square metres. Thus the results indicate that the manufacturer is

    exaggerating the area covered.

     If a one-tailed test was appropriate, the alternative in Options would be set to

    less than or more than. Note that a confidence interval is only given when a

    two-tailed alternative is used.

    3. Go back to the dialog box for a t-test and select the graph option to obtain the following boxplot. The red line indicates the 95% confidence interval and the blue dot is at14, the mean stated in the null hypothesis.

    Boxplot of area

    (with Ho and 95% t-confidence interval for the mean)

    []_XHo

    12.513.514.5

    area

    4. The t-test is based on the assumption that the data follow a normal distribution. One way of testing this is to obtain a normal probability plot of the data. Select

    Stat>Basic statistics>Normality test

    Enter area in the Variable box. You should obtain the following plot. The fact that the plot is approximately linear indicates the assumption of normality is valid.

    Normal Probability Plot

    .999

    .99

    .95

    .80

    .50

    .20Probability.05

    .01

    .001

    12.513.514.5

    areaAverage: 13.3917Anderson-Darling Normality TestStDev: 0.665321A-Squared: 0.235N: 12P-Value: 0.732

    Example 1

    Established records show that the lengths of mussels from an estuary are normally distributed with a mean of 30 mm. The following data are lengths from a random sample of 25 mussels taken from a polluted beach. It is suspected that the effect of the pollution will be to inhibit the growth of the mussels.

    27.72 17.44 19.72 42.39 22.31

    30.87 20.06 18.03 16.29 24.95

    19.15 32.22 27.33 35.88 18.57

    22.02 27.45 26.56 22.32 31.40

    19.12 43.56 40.63 36.12 26.95

    a) Write down appropriate null and alternative hypotheses for this example. b) Enter the data into Minitab and carry out the appropriate test. From the results

    comment on whether or not pollution appears to inhibit the growth of mussels. c) Obtain each of the graphs available in the t-test options and make sure you

    understand the information given on these.

    d) Obtain a normal plot to test the assumption of normality

    e) Obtain a 95% confidence interval for the mean percentage length of the mussels

    from the polluted beach.

Example 2

    The angles, measured in degrees, between the first two segments of fifteen hyphae of a certain fungus were as follows:

     115 124 126 121 135 113 119 116 116 112 123 122 130 113 134

    a) A researcher has postulated that the mean angle should be 120 degrees.

    Carry out a t-test, using a 5% significance level, to assess whether the data

    are compatible with this claim.

    b) What assumption have you made and is it valid?

Example 3

    The weights (g) of eight adult starlings caught at a roost were as follows:

    78 82 88 81 87 80 88 80

    a) Calculate a 95% confidence interval for the mean weight of starlings in the

    roost. (Use the one-sample t dialog box entering any value for the mean.) b) Calculate a 99% confidence interval for the mean weight. (Go to options in the

    one-sample t dialog box and change 95 to 99.) How do the intervals compare. (c) What assumptions are you making to calculate these confidence intervals?

    Check that this is valid.

     __________________________________________________________________

     Hypothesis testing: Paired and two-sample tests The manufacturer of a suntan lotion wants to know whether or not a new ingredient increases the protection against sunburn. Seven volunteers have their backs exposed to a sun lamp with the old lotion on one side and the new lotion on the other side of the spine. A higher number indicates more burning. Does the new ingredient improve the effectiveness of the lotion? The data are:

    volunteer 1 2 3 4 5 6 7

    burn without new ingredient 42 51 31 61 44 55 48