The following histogram was obtained by counting the...

The following histogram was obtained by counting the occurrence of the first digits in telephone numbers in one column of a telephone directory: Table can't copy Test the goodness of fit of this data to a random variable that is uniformly distributed in the set $\{0,1, \ldots, 9\}$ at a $1 \%$ significance level. Repeat for the set $\{2,3, \ldots, 9\}$.

The following histogram was obtained by counting the occurrence of the first digits in telephone numbers in one column of a telephone directory:
Table can't copy
Test the goodness of fit of this data to a random variable that is uniformly distributed in the set $\{0,1, \ldots, 9\}$ at a $1 \%$ significance level. Repeat for the set $\{2,3, \ldots, 9\}$.

Key Concepts

Degrees of Freedom

A parameter that reflects the number of independent values or categories in the data that are free to vary. In a chi-square goodness-of-fit test, it is typically calculated as the number of categories minus one, which directly influences the comparison with the chi-square distribution.

Null Hypothesis

The assumption made for the purposes of testing, stating that there is no significant difference between the observed and expected frequencies—that is, the data follow the hypothesized uniform distribution.

Significance Level

A predefined threshold (in this case, 1%) used to assess whether the observed deviations from the expected frequencies are statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true.

Chi-Square Test

A commonly used method for performing a goodness-of-fit test, where the test statistic is calculated as the sum of the squared differences between observed and expected counts, divided by the expected counts. The magnitude of this statistic, when compared against the chi-square distribution with the appropriate degrees of freedom, helps decide whether to reject the null hypothesis.

Goodness-of-Fit Test

A statistical method used to determine how well observed data match the data expected under a specific hypothesis. It involves comparing the observed frequency distribution of data to a theoretical distribution to decide if any deviations are due to random chance or indicate a significant difference.

Uniform Distribution

A probability distribution in which all outcomes are equally likely. In the context of this problem, it is hypothesized that the first digits occur with equal frequency, meaning each digit has the same probability of appearing.

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