Question

after peering into a wishing well, jocelyn hypothesizes that the distribution of coins is 50% quarters, 20% dimes, 20% nickels, and 10% pennies. to investigate this claim, she reaches in, mixes the coins well, and selects a random sample of 80 coins. she counts how many coins there are of each type. she would like to know if her observed distribution provides convincing evidence that the true distribution of coins in the well is different from what she originally thought. what are the appropriate hypotheses?

$h_{0}:p_{quarter}=0.5,p_{dime}=0.2,p_{nickel}=0.2,p_{penny}=0.1$
$h_{a}$: not all of the $p_{i}$s are as stated.

$h_{0}$: not all of the $p_{i}$s are as stated.
$h_{a}:p_{quarter}=0.5,p_{dime}=0.2,p_{nickel}=0.2,p_{penny}=0.1$

$h_{0}:p_{quarter}=40,p_{dime}=16,p_{nickel}=16,p_{penny}=8$
$h_{a}:p_{quarter}=40,p_{dime}=16,p_{nickel}=16,p_{penny}=8$

$h_{0}:p_{quarter}=40,p_{dime}=16,p_{nickel}=16,p_{penny}=8$
$h_{a}:p_{quarter}
eq40,p_{dime}
eq16,p_{nickel}
eq16,p_{penny}
eq8$

Explanation:

Step1: Recall the null and alternative hypotheses for a chi - square goodness - of - fit test

The null hypothesis \(H_0\) for a chi - square goodness - of - fit test states that the distribution of the categorical variable (in this case, the proportion of different coins) is as hypothesized. The alternative hypothesis \(H_a\) states that the distribution is not as hypothesized.

For the coin problem, the hypothesized proportions are \(p_{quarter}=0.5\), \(p_{dime}=0.2\), \(p_{nickel}=0.2\), \(p_{penny}=0.1\).

The null hypothesis \(H_0\) is \(p_{quarter}=0.5\), \(p_{dime}=0.2\), \(p_{nickel}=0.2\), \(p_{penny}=0.1\)

The alternative hypothesis \(H_a\) is that not all of the \(p_i\)s are as stated.

Step2: Analyze each option

• Option 1:

\(H_0:p_{quarter}=0.5,p_{dime}=0.2,p_{nickel}=0.2,p_{penny}=0.1\)
\(H_a:\text{Not all of the }p_i\text{s are as stated}\)
This is the correct form of the null and alternative hypotheses for a chi - square goodness - of - fit test.

• Option 2:

The null and alternative hypotheses are reversed. The null hypothesis should be the statement of the hypothesized distribution, not the alternative.

• Option 3 and 4:

The hypotheses are stated in terms of counts (\(n = 80\), so \(np_{quarter}=80\times0.5 = 40\), \(np_{dime}=80\times0.2=16\), \(np_{nickel}=80\times0.2 = 16\), \(np_{penny}=80\times0.1=8\)). But in a chi - square goodness - of - fit test, the hypotheses are about proportions (\(p\) values), not counts.

Answer:

\(H_0:p_{quarter}=0.5,p_{dime}=0.2,p_{nickel}=0.2,p_{penny}=0.1\)
\(H_a:\text{Not all of the }p_i\text{s are as stated}\) (corresponding to the first option in the multiple - choice list)