QuestionSurvey skiers' preferences for ski areas. Test if preference is independent of skill level at $\alpha=0.05$ . Find test statistic and $p$ -value.

Studdy Solution

STEP 1

Assumptions1. The survey results are represented in the given table. . The null hypothesis $H_{0}$ is that the best ski area is independent of the level of the skier.
3. The alternative hypothesis $H_{1}$ is that the best ski area is not independent of the level of the skier.
4. The significance level $\alpha$ is0.05.

STEP 2

First, we need to calculate the row totals, column totals, and grand total of the survey results.

STEP 3

The row totals are calculated by adding up the numbers in each row.
$ahoe\, total =20 +30 +41 =91$ $Utah\, total =10 +29 +62 =101$ $Colorado\, total =10 +38 +49 =97$

STEP 4

The column totals are calculated by adding up the numbers in each column.
$Beginner\, total =20 +10 +10 =40$ $Intermediate\, total =30 +29 +38 =97$ $Advanced\, total =41 +62 +49 =152$

STEP 5

The grand total is calculated by adding up all the numbers in the table.
$Grand\, total =91 +101 +97 =289$

STEP 6

Next, we need to calculate the expected frequencies for each cell in the table. The expected frequency is calculated by multiplying the row total and column total for that cell, and then dividing by the grand total.
$Expected\, frequency = \frac{Row\, total \\times Column\, total}{Grand\, total}$

STEP 7

Calculate the expected frequencies for each cell in the table.

STEP 8

Now, we can calculate the test statistic, which is a chi-square statistic. The chi-square statistic is calculated by summing the squared differences between the observed and expected frequencies, divided by the expected frequency, for all cells in the table.
$\chi^2 = \sum \frac{(Observed\, frequency - Expected\, frequency)^2}{Expected\, frequency}$

STEP 9

Calculate the chi-square statistic.

STEP 10

The degrees of freedom for the chi-square test is calculated as $(r-)(c-)$ , where $r$ is the number of rows and $c$ is the number of columns in the table.
$df = (r-)(c-)$

STEP 11

Calculate the degrees of freedom.

STEP 12

Now, we can find the p-value associated with the calculated chi-square statistic and degrees of freedom. The p-value is the probability of obtaining a chi-square statistic as extreme as, or more extreme than, the calculated value, under the null hypothesis.

STEP 13

Calculate the p-value.

STEP 14

Compare the p-value to the significance level $\alpha$ . If the p-value is less than or equal to $\alpha$ , we reject the null hypothesis. If the p-value is greater than $\alpha$ , we do not reject the null hypothesis.

STEP 15

Make the decision to reject or not reject the null hypothesis.

STEP 16

Write the conclusion based on the decision made in the previous step.

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