Question

the table below shows the distribution of birth days in a random sample of 148 births from local boston records:
day of birth | number of babies born that day
--- | ---
sun | 21
mon | 16
tues | 26
wed | 26
thurs | 16
fri | 26
sat | 17

are births evenly distributed across the days of the week?
we will use a hypothesis to help decide, below. (let’s use a 0.05 level of significance.)

step 1. identify hypotheses ( h_0 ) and ( h_a ):
what type of test will we be doing? select an answer

( h_0 ): select an answer
( h_a ): select an answer

question help: video 1, video 2, written example 1

Explanation:

Step1: Determine Test Type

We are testing if births are evenly distributed across days of the week, which is a goodness - of - fit test for a multinomial distribution (specifically a chi - square goodness - of - fit test). For a chi - square goodness - of - fit test, we check if the observed frequencies match the expected frequencies under a specified distribution (here, uniform distribution across 7 days).

Step2: State Null Hypothesis ($H_0$)

The null hypothesis for a goodness - of - fit test in this case is that births are evenly distributed across the days of the week. Mathematically, if $p_i$ is the proportion of births on day $i$ ($i = 1,2,\cdots,7$ for the 7 days of the week), then $H_0:p_1 = p_2=\cdots=p_7=\frac{1}{7}$, which means the number of babies born each day is the same (expected frequency for each day is $\frac{148}{7}\approx21.14$).

Step3: State Alternative Hypothesis ($H_A$)

The alternative hypothesis is that births are not evenly distributed across the days of the week. So, at least one of the $p_i$ is different from $\frac{1}{7}$, i.e., the number of babies born each day is not the same.

Answer:

• Type of test: Chi - square goodness - of - fit test
• $H_0$: Births are evenly distributed across the days of the week (or the proportion of births on each day of the week is equal, $p_1 = p_2=\cdots=p_7=\frac{1}{7}$)
• $H_A$: Births are not evenly distributed across the days of the week (or at least one of the proportions $p_i$ is different from $\frac{1}{7}$)