Question

a company is started by four friends. the company was ericas idea, so she wants to fill 70% of the orders. jen, heather, and tonya each agree to fill 10% of the orders. after a successful first year, erica wants to determine if the distribution of the number of orders filled is adhering to the agreed - upon percentages. to do so, she selects a random sample of 100 orders from the large number of orders that were filled and determines who filled the order. what are the appropriate hypotheses for this test?

○ ( h_0:p_{\text{erica}} = 0.7,p_{\text{jen}} = 0.1,p_{\text{heather}} = 0.1,p_{\text{tonya}} = 0.1 )
( h_a:p_{\text{erica}}
eq0.7,p_{\text{jen}}
eq0.1,p_{\text{heather}}
eq0.1,p_{\text{tonya}}
eq0.1 )

○ ( h_0:p_{\text{erica}}
eq0.7,p_{\text{jen}}
eq0.1,p_{\text{heather}}
eq0.1,p_{\text{tonya}}
eq0.1 )
( h_a:p_{\text{erica}} = 0.7,p_{\text{jen}} = 0.1,p_{\text{heather}} = 0.1,p_{\text{tonya}} = 0.1 )

○ ( h_0:p_{\text{erica}} = 0.7,p_{\text{jen}} = 0.1,p_{\text{heather}} = 0.1,p_{\text{tonya}} = 0.1 )
( h_a:\text{not all of the }p_i\text{s are as stated.} )

○ ( h_0:\text{not all of the }p_i\text{s are as stated.} )
( h_a:p_{\text{erica}} = 0.7,p_{\text{jen}} = 0.1,p_{\text{heather}} = 0.1,p_{\text{tonya}} = 0.1 )

Explanation:

In a chi - square goodness - of - fit test (which is appropriate here as we are checking if the observed distribution of orders adheres to a hypothesized distribution), the null hypothesis \(H_0\) states that the proportions are as specified. The alternative hypothesis \(H_a\) for a goodness - of - fit test is that the proportions are not all as specified.
The first option with \(H_a\) having \(p_{Erica}
eq0.7\), \(p_{Jen}
eq0.1\), \(p_{Heather}
eq0.1\), \(p_{Tonya}
eq0.1\) is incorrect because it implies all proportions are different simultaneously in a very strict (and not the general) way. The second option has the null and alternative hypotheses reversed. The fourth option has the null and alternative hypotheses reversed.

Answer:

\(H_0:p_{Erica} = 0.7,p_{Jen}=0.1,p_{Heather}=0.1,p_{Tonya}=0.1\) and \(H_a:\text{Not all of the }p_i\text{'s are as stated}\) (the third option).