Question

a random sample of 772 subjects was asked to identify the day of the week that is best for quality family time. consider the claim that the days of the week are selected with a uniform distribution so that all days have the same chance of being selected. the table below shows goodness - of - fit test results from the claim and data from the study. test that claim using either the critical value method or the p - value method with an assumed significance level of $\alpha = 0.05$.
num categories 7
degrees of freedom 6
expected freq 110.2857
test statistic, $\chi^{2}$ 10.462
critical $\chi^{2}$ 12.592
p - value 0.1065
determine the null and alternative hypotheses.
$h_{0}$:
$h_{1}$:
at least two days of the week have a different frequency of being selected.
all days of the week have a different chance of being selected.
all days of the week have an equal chance of being selected.
at least one day of the week has a different chance of being selected.

Explanation:

Step1: Define Null Hypothesis ($H_0$)

The claim is that the days of the week are selected with a uniform distribution (all days have the same chance). So, the null hypothesis states that all days of the week have an equal chance of being selected.

Step2: Define Alternative Hypothesis ($H_1$)

The alternative hypothesis is the opposite of the null hypothesis. If the null is that all days have equal chance, the alternative is that at least one day has a different chance of being selected.

Answer:

$H_0$: All days of the week have an equal chance of being selected.
$H_1$: At least one day of the week has a different chance of being selected.