Answer: Let the hypotheses be:
\(H_0\): shell size is independent of beach location.
\(H_1\): shell size is not independent of beach location.
The expected frequencies are:
| A | B | C | D |
|---|
| Large | \(60.29\) | \(77.87\) | \(100.48\) | \(75.36\) |
|---|
| Small | \(35.71\) | \(46.13\) | \(59.52\) | \(44.64\) |
|---|
The chi-squared test statistic is \(\chi^2 = 7.04\).
Degrees of freedom: \((2-1)(4-1)=3\).
At the \(10\%\) significance level, the critical value for \(3\) degrees of freedom is \(6.251\).
Since \(7.04 \gt 6.251\), reject \(H_0\).
There is sufficient evidence to suggest that shell size depends on beach location, so shell size is not independent of location.
We use a chi-squared test of independence.
Set up the hypotheses:
\(H_0\): shell size is independent of beach location.
\(H_1\): shell size is not independent of beach location.
The observed frequencies are:
| A | B | C | D | Total |
|---|
| Large | 68 | 69 | 96 | 81 | 314 |
|---|
| Small | 28 | 55 | 64 | 39 | 186 |
|---|
| Total | 96 | 124 | 160 | 120 | 500 |
|---|
If size and location are independent, each expected frequency is
\(\text{Expected} = \dfrac{(\text{row total})(\text{column total})}{\text{grand total}}\).
So:
- Large at A: \(\dfrac{314\times 96}{500}=60.288\)
- Large at B: \(\dfrac{314\times 124}{500}=77.872\)
- Large at C: \(\dfrac{314\times 160}{500}=100.48\)
- Large at D: \(\dfrac{314\times 120}{500}=75.36\)
- Small at A: \(\dfrac{186\times 96}{500}=35.712\)
- Small at B: \(\dfrac{186\times 124}{500}=46.128\)
- Small at C: \(\dfrac{186\times 160}{500}=59.52\)
- Small at D: \(\dfrac{186\times 120}{500}=44.64\)
So the expected frequency table is:
| A | B | C | D |
|---|
| Large | \(60.29\) | \(77.87\) | \(100.48\) | \(75.36\) |
|---|
| Small | \(35.71\) | \(46.13\) | \(59.52\) | \(44.64\) |
|---|
Now calculate \(\dfrac{(O-E)^2}{E}\) for each cell:
| A | B | C | D |
|---|
| Large | \(\dfrac{(68-60.288)^2}{60.288}=0.9859\) | \(\dfrac{(69-77.872)^2}{77.872}=1.0104\) | \(\dfrac{(96-100.48)^2}{100.48}=0.1997\) | \(\dfrac{(81-75.36)^2}{75.36}=0.4221\) |
|---|
| Small | \(\dfrac{(28-35.712)^2}{35.712}=1.6646\) | \(\dfrac{(55-46.128)^2}{46.128}=1.7055\) | \(\dfrac{(64-59.52)^2}{59.52}=0.3372\) | \(\dfrac{(39-44.64)^2}{44.64}=0.7126\) |
|---|
Hence
\(\chi^2 = 0.9859+1.0104+0.1997+0.4221+1.6646+1.7055+0.3372+0.7126 = 7.0379\).
So the test statistic is \(\chi^2 \approx 7.04\).
The number of degrees of freedom is
\((2-1)(4-1)=3\).
At the \(10\%\) significance level, the critical value for \(3\) degrees of freedom is \(6.251\).
Since \(7.04 \gt 6.251\), we reject \(H_0\).
There is sufficient evidence at the \(10\%\) level to suggest that shell size is dependent on beach location.
