Math  /  Data & Statistics

QuestionYou are conducting a test of the claim that the color of the case of your cell phone is dependent on your gender. \begin{tabular}{|c||c|c|c|} \hline & Black & White & Blue \\ \hline Male & 80 & 88 & 51 \\ \hline Female & 56 & 14 & 94 \\ \hline \hline \end{tabular}
Give all answers rounded to 2 places after the decimal point, if necessary. (a) What is the chi-square test-statistic for this data?
Test Statistic: χ2=\chi^{2}= \square (b) What is the critical value for this test of independence when using a significance level of α=0.05\alpha=0.05 ? Critical Value: χ2=\chi^{2}= \square (c) What is the correct summary of this hypothesis test at the 0.05 significance level? I was unable to show that your choice of color is dependent on your gender. I was able to show that your choice of color is independent of your gender. I was unable to show that your choice of color is independent of your gender. I was able to show that your choice of color is dependent on your gender. Question Help: Fig Written Example
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Studdy Solution

STEP 1

1. We are using a chi-square test of independence.
2. The data is presented in a contingency table format.
3. The significance level α\alpha is 0.05.

STEP 2

1. Calculate the expected frequencies.
2. Compute the chi-square test statistic.
3. Determine the degrees of freedom.
4. Find the critical value for the chi-square distribution.
5. Compare the test statistic to the critical value and summarize the hypothesis test.

STEP 3

Calculate the row totals, column totals, and grand total.
Row totals: - Male: 80+88+51=21980 + 88 + 51 = 219 - Female: 56+14+94=16456 + 14 + 94 = 164
Column totals: - Black: 80+56=13680 + 56 = 136 - White: 88+14=10288 + 14 = 102 - Blue: 51+94=14551 + 94 = 145
Grand total: 219+164=383219 + 164 = 383

STEP 4

Calculate the expected frequencies using the formula: E=(Row Total×Column Total)Grand Total E = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}
Expected frequencies: - EMale, Black=219×13638377.73E_{\text{Male, Black}} = \frac{219 \times 136}{383} \approx 77.73 - EMale, White=219×10238358.34E_{\text{Male, White}} = \frac{219 \times 102}{383} \approx 58.34 - EMale, Blue=219×14538382.93E_{\text{Male, Blue}} = \frac{219 \times 145}{383} \approx 82.93 - EFemale, Black=164×13638358.27E_{\text{Female, Black}} = \frac{164 \times 136}{383} \approx 58.27 - EFemale, White=164×10238343.66E_{\text{Female, White}} = \frac{164 \times 102}{383} \approx 43.66 - EFemale, Blue=164×14538362.07E_{\text{Female, Blue}} = \frac{164 \times 145}{383} \approx 62.07

STEP 5

Compute the chi-square test statistic using the formula: χ2=(OE)2E \chi^2 = \sum \frac{(O - E)^2}{E}
Where OO is the observed frequency and EE is the expected frequency.
χ2=(8077.73)277.73+(8858.34)258.34+(5182.93)282.93+(5658.27)258.27+(1443.66)243.66+(9462.07)262.07\chi^2 = \frac{(80 - 77.73)^2}{77.73} + \frac{(88 - 58.34)^2}{58.34} + \frac{(51 - 82.93)^2}{82.93} + \frac{(56 - 58.27)^2}{58.27} + \frac{(14 - 43.66)^2}{43.66} + \frac{(94 - 62.07)^2}{62.07}
χ2(2.27)277.73+(29.66)258.34+(31.93)282.93+(2.27)258.27+(29.66)243.66+(31.93)262.07\chi^2 \approx \frac{(2.27)^2}{77.73} + \frac{(29.66)^2}{58.34} + \frac{(-31.93)^2}{82.93} + \frac{(-2.27)^2}{58.27} + \frac{(-29.66)^2}{43.66} + \frac{(31.93)^2}{62.07}
χ20.07+15.07+12.29+0.09+20.14+16.42\chi^2 \approx 0.07 + 15.07 + 12.29 + 0.09 + 20.14 + 16.42
χ264.08\chi^2 \approx 64.08

STEP 6

Determine the degrees of freedom using the formula: df=(number of rows1)×(number of columns1) \text{df} = (\text{number of rows} - 1) \times (\text{number of columns} - 1)
df=(21)×(31)=2\text{df} = (2 - 1) \times (3 - 1) = 2

STEP 7

Find the critical value for α=0.05\alpha = 0.05 and df=2\text{df} = 2 using a chi-square distribution table.
Critical value χ0.05,225.99\chi^2_{0.05, 2} \approx 5.99

STEP 8

Compare the test statistic to the critical value.
Since χ264.08\chi^2 \approx 64.08 is greater than the critical value 5.995.99, we reject the null hypothesis.
Summary: The correct summary of this hypothesis test at the 0.05 significance level is: "I was able to show that your choice of color is dependent on your gender."
Test Statistic: χ264.08\chi^2 \approx 64.08
Critical Value: χ25.99\chi^2 \approx 5.99

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