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Sagot :
To determine whether the observed frequencies of customers served align with the expected uniform distribution, we need to perform a Chi-Square goodness-of-fit test. Let's go through the steps in detail:
1. Observed Frequencies:
- The number of customers served each day are as follows:
[tex]\[ \text{Monday: } 40, \quad \text{Tuesday: } 33, \quad \text{Wednesday: } 35, \quad \text{Thursday: } 32, \quad \text{Friday: } 60 \][/tex]
This gives us the observed frequencies:
[tex]\[ \mathbf{Observed} = [40, 33, 35, 32, 60] \][/tex]
2. Total Number of Observations:
- The total number of customers served in the week is:
[tex]\[ \text{Total} = 200 \][/tex]
3. Expected Frequencies:
- If the customers are served uniformly throughout the week, we expect the same number of customers each day. Given there are 5 days, the expected frequency for each day can be calculated as:
[tex]\[ \text{Expected Frequency} = \frac{\text{Total}}{5} = \frac{200}{5} = 40 \][/tex]
This gives us the expected frequencies:
[tex]\[ \mathbf{Expected} = [40, 40, 40, 40, 40] \][/tex]
4. Calculate the Chi-Square Test Statistic:
- The Chi-Square test statistic [tex]\(\chi^2\)[/tex] is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}} \][/tex]
Substituting the observed and expected frequencies into this formula, we get:
[tex]\[ \chi^2 = \frac{(40 - 40)^2}{40} + \frac{(33 - 40)^2}{40} + \frac{(35 - 40)^2}{40} + \frac{(32 - 40)^2}{40} + \frac{(60 - 40)^2}{40} \][/tex]
Simplifying each term, we get:
[tex]\[ \frac{(40 - 40)^2}{40} = \frac{0^2}{40} = 0 \][/tex]
[tex]\[ \frac{(33 - 40)^2}{40} = \frac{(-7)^2}{40} = \frac{49}{40} = 1.225 \][/tex]
[tex]\[ \frac{(35 - 40)^2}{40} = \frac{(-5)^2}{40} = \frac{25}{40} = 0.625 \][/tex]
[tex]\[ \frac{(32 - 40)^2}{40} = \frac{(-8)^2}{40} = \frac{64}{40} = 1.6 \][/tex]
[tex]\[ \frac{(60 - 40)^2}{40} = \frac{20^2}{40} = \frac{400}{40} = 10 \][/tex]
Summing these results:
[tex]\[ \chi^2 = 0 + 1.225 + 0.625 + 1.6 + 10 = 13.45 \][/tex]
Therefore, the test statistic [tex]\(\chi^2\)[/tex] is [tex]\(13.45\)[/tex].
1. Observed Frequencies:
- The number of customers served each day are as follows:
[tex]\[ \text{Monday: } 40, \quad \text{Tuesday: } 33, \quad \text{Wednesday: } 35, \quad \text{Thursday: } 32, \quad \text{Friday: } 60 \][/tex]
This gives us the observed frequencies:
[tex]\[ \mathbf{Observed} = [40, 33, 35, 32, 60] \][/tex]
2. Total Number of Observations:
- The total number of customers served in the week is:
[tex]\[ \text{Total} = 200 \][/tex]
3. Expected Frequencies:
- If the customers are served uniformly throughout the week, we expect the same number of customers each day. Given there are 5 days, the expected frequency for each day can be calculated as:
[tex]\[ \text{Expected Frequency} = \frac{\text{Total}}{5} = \frac{200}{5} = 40 \][/tex]
This gives us the expected frequencies:
[tex]\[ \mathbf{Expected} = [40, 40, 40, 40, 40] \][/tex]
4. Calculate the Chi-Square Test Statistic:
- The Chi-Square test statistic [tex]\(\chi^2\)[/tex] is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}} \][/tex]
Substituting the observed and expected frequencies into this formula, we get:
[tex]\[ \chi^2 = \frac{(40 - 40)^2}{40} + \frac{(33 - 40)^2}{40} + \frac{(35 - 40)^2}{40} + \frac{(32 - 40)^2}{40} + \frac{(60 - 40)^2}{40} \][/tex]
Simplifying each term, we get:
[tex]\[ \frac{(40 - 40)^2}{40} = \frac{0^2}{40} = 0 \][/tex]
[tex]\[ \frac{(33 - 40)^2}{40} = \frac{(-7)^2}{40} = \frac{49}{40} = 1.225 \][/tex]
[tex]\[ \frac{(35 - 40)^2}{40} = \frac{(-5)^2}{40} = \frac{25}{40} = 0.625 \][/tex]
[tex]\[ \frac{(32 - 40)^2}{40} = \frac{(-8)^2}{40} = \frac{64}{40} = 1.6 \][/tex]
[tex]\[ \frac{(60 - 40)^2}{40} = \frac{20^2}{40} = \frac{400}{40} = 10 \][/tex]
Summing these results:
[tex]\[ \chi^2 = 0 + 1.225 + 0.625 + 1.6 + 10 = 13.45 \][/tex]
Therefore, the test statistic [tex]\(\chi^2\)[/tex] is [tex]\(13.45\)[/tex].
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