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Many businesses have music piped into the work areas to improve the environment. At a company an experiment is performed to compare different types of music. Three types of music – country, rock and classical – are tried each on four randomly selected day. Each day the productivity, measured by the number of items produced, is recorded. The results appear below: / Baie besighede het musiek ingebring in die werkplek om die omgewing te verbeter. By ‘n besigheid is ‘n ekspriment uitgevoer om die verskillende tiepes musiek te vergelyk. Drie tipes musiek – country, rok en klassiek – word gespeel op vier ewekansige gekose dae. Elke dag word die produktiwiteit gemeet deur die aantal items wat geproduseer word. Die resultate verskyn onder:

Country Rock Classical
857 791 824
801 753 847
795 781 881
842 776 865


Can we conclude from this information that the mean number of items produced differs for at least two of the three types of music? / Kan ons tot die gevolgtrekking uit hierdie informasie kom dat die gemiddelde aantal items verskil vir tenminste twee van die drie tiepes musiek? Test at 5% significance level. Calculate SSTotal. / Toets by ‘n 5% betekenispeil. Bereken SSTotal.

1 answer

To test whether the mean number of items produced differs for at least two of the three types of music, we can perform a one-way ANOVA test. Before we conduct the ANOVA test, we need to calculate the total sum of squares (\(SSTotal\)).

The total sum of squares (\(SSTotal\)) is calculated in the following steps:

  1. Calculate the overall mean (\(\bar{X}_{total}\)) of all observations.
  2. Calculate the total sum of squares (\(SSTotal\)), which is the sum of the squares of the differences between each observation and the overall mean.

Step 1: Calculate the Overall Mean

We first sum all the observations and then divide by the total number of observations.

The data is as follows:

  • Country: 857, 801, 795, 842
  • Rock: 791, 753, 781, 776
  • Classical: 824, 847, 881, 865

Let’s calculate the total and the overall mean:

\[ \text{Total} = 857 + 801 + 795 + 842 + 791 + 753 + 781 + 776 + 824 + 847 + 881 + 865 \] \[ \text{Total} = 9,365 \]

There are a total of \(12\) observations (4 for each type of music).

\[ \bar{X}_{total} = \frac{9365}{12} = 780.417 \]

Step 2: Calculate the Total Sum of Squares (\(SSTotal\))

The total sum of squares is calculated as follows:

\[ SSTotal = \sum (X_{ij} - \bar{X}_{total})^2 \]

Where \(X_{ij}\) are the individual observations.

Now, calculate each squared difference and sum them:

  1. For Country:

    • \( (857 - 780.417)^2 = (76.583)^2 = 5854.134 \)
    • \( (801 - 780.417)^2 = (20.583)^2 = 423.536 \)
    • \( (795 - 780.417)^2 = (14.583)^2 = 212.673 \)
    • \( (842 - 780.417)^2 = (61.583)^2 = 3792.616 \)

    Total for Country: \[ 5854.134 + 423.536 + 212.673 + 3792.616 = 10,283.959 \]

  2. For Rock:

    • \( (791 - 780.417)^2 = (10.583)^2 = 112.022 \)
    • \( (753 - 780.417)^2 = (27.417)^2 = 752.597 \)
    • \( (781 - 780.417)^2 = (0.583)^2 = 0.340 \)
    • \( (776 - 780.417)^2 = (4.417)^2 = 19.556 \)

    Total for Rock: \[ 112.022 + 752.597 + 0.340 + 19.556 = 884.515 \]

  3. For Classical:

    • \( (824 - 780.417)^2 = (43.583)^2 = 1895.411 \)
    • \( (847 - 780.417)^2 = (66.583)^2 = 4422.865 \)
    • \( (881 - 780.417)^2 = (100.583)^2 = 10116.563 \)
    • \( (865 - 780.417)^2 = (84.583)^2 = 7167.274 \)

    Total for Classical: \[ 1895.411 + 4422.865 + 10116.563 + 7167.274 = 23602.113 \]

Total \(SSTotal\)

\[ SSTotal = 10,283.959 + 884.515 + 23602.113 = 37641.587 \]

Thus, the total sum of squares, \(SSTotal\), is approximately 37641.59 (rounded to two decimal places).

Next, you would compare this alongside the between-group and within-group sums of squares and perform the ANOVA test to ultimately conclude whether there's a significant difference in productivity based on the type of music played, but that would involve further calculations and interpretations.