Question

A research group wants to determine whether the proportion of car accidents that were caused by drivers using cell phones has changed from the previous value of 13%. They obtained 10,000 auto accident reports and found that 14% were caused by drivers using cell phones. Find the test statistic.

Answer 1

The test statistic, z, is approximately 1.154. This value indicates a slight but potentially non-significant increase in the proportion of cell phone-related accidents compared to the previous value of 13%.

Step-by-step explanation:

Null and Alternative Hypotheses:

Null Hypothesis (H0): The proportion of accidents caused by cell phones has not changed, p = 0.13.

Alternative Hypothesis (Ha): The proportion has changed, p ≠ 0.13.

Test Statistic:

We can use the z-test for proportions to calculate the test statistic.

z = (Observed proportion - Expected proportion) / Standard Error

Observed proportion = 0.14 (14% from the sample)

Expected proportion = 0.13 (previous value)

Standard Error = sqrt(p * (1-p) / n) ≈ sqrt(0.13 * 0.87 / 10,000) ≈ 0.003

Calculation:

z = (0.14 - 0.13) / 0.003 ≈ 1.154

Therefore, the z-statistic is approximately 1.154.

Interpretation:

A z-score closer to 0 indicates no evidence against the null hypothesis (no change). Higher positive or negative values suggest increasing evidence for the alternative hypothesis (change). In this case, z = 1.154 is slightly positive, suggesting a potential but not conclusive increase in the proportion of cell phone-related accidents. Further analysis, such as p-value calculation, is needed to determine the statistical significance of this difference.

Answer 2

Answer: -2.88

Step-by-step explanation:

Hypothesis : The proportion of car accidents that were caused by drivers using cell phones has changed from the previous value =13%

i.e. p=0.13

Sample size :
`10,000`

The proportion of auto accident were caused by drivers using cell phones.

`P=0.14`

Now, the test statistic for the population proportion is given by :-

`z=\frac{p-P}{\sqrt{(P(1-P))/(n)}}`

`\Rightarrow\ z=\frac{0.13-0.14}{\sqrt{(0.14(1-0.14))/(10000)}}\Rightarrow\ z=-2.881952088\approx-2.88`

Hence, the value of the test statistic is -2.88