The history of 100 workers who lost their employment due to technological advances is reviewed. Each worker was given an alterna
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The required proof is given in the table below:
![\begin{tabular}{|p{4cm}|p{6cm}|} Statement & Reason \\ [1ex] 1. $\overline{BD}$ bisects $\angle ABC$ & 1. Given \\ 2. \angle DBC\cong\angle ABD & 2. De(finition of angle bisector \\ 3. $\overline{AE}$||$\overline{BD}$ & 3. Given \\ 4. \angle AEB\cong\angle DBC & 4. Corresponding angles \\ 5. \angle AEB\cong\angle ABD & 5. Transitive property of equality \\ 6. \angle ABD\cong\angle BAE & 6. Alternate angles \end{tabular}](https://tex.z-dn.net/?f=%20%5Cbegin%7Btabular%7D%7B%7Cp%7B4cm%7D%7Cp%7B6cm%7D%7C%7D%20%0A%20Statement%20%26%20Reason%20%5C%5C%20%5B1ex%5D%20%0A1.%20%24%5Coverline%7BBD%7D%24%20bisects%20%24%5Cangle%20ABC%24%20%26%201.%20Given%20%5C%5C%0A2.%20%5Cangle%20DBC%5Ccong%5Cangle%20ABD%20%26%202.%20De%28finition%20of%20angle%20bisector%20%5C%5C%20%0A3.%20%24%5Coverline%7BAE%7D%24%7C%7C%24%5Coverline%7BBD%7D%24%20%26%203.%20Given%20%5C%5C%20%0A4.%20%5Cangle%20AEB%5Ccong%5Cangle%20DBC%20%26%204.%20Corresponding%20angles%20%5C%5C%0A5.%20%5Cangle%20AEB%5Ccong%5Cangle%20ABD%20%26%205.%20Transitive%20property%20of%20equality%20%5C%5C%20%0A6.%20%5Cangle%20ABD%5Ccong%5Cangle%20BAE%20%26%206.%20Alternate%20angles%0A%5Cend%7Btabular%7D)
![\begin{tabular}{|p{4cm}|p{6cm}|} 7. \angle AEB\cong\angle BAE & 7. Transitive property of equality \\ 8. \overline{EB}\cong\overline{AB} & 8. From 7. $\Delta ABE$ is isosceles \\ 9. EB = AB & 9. De(finition of congruence \\ 10. $\frac{AD}{DC}=\frac{EB}{BC}$ & 10. Triangle proportionality theorem \\ 11. $\frac{AD}{DC}=\frac{AB}{BC}$ & 11. Substitution Property of equality \\[1ex] \end{tabular} ](https://tex.z-dn.net/?f=%5Cbegin%7Btabular%7D%7B%7Cp%7B4cm%7D%7Cp%7B6cm%7D%7C%7D%0A7.%20%5Cangle%20AEB%5Ccong%5Cangle%20BAE%20%26%207.%20Transitive%20property%20of%20equality%20%5C%5C%0A8.%20%5Coverline%7BEB%7D%5Ccong%5Coverline%7BAB%7D%20%26%208.%20From%207.%20%24%5CDelta%20ABE%24%20is%20isosceles%20%5C%5C%0A9.%20EB%20%3D%20AB%20%26%209.%20De%28finition%20of%20congruence%20%5C%5C%2010.%20%24%5Cfrac%7BAD%7D%7BDC%7D%3D%5Cfrac%7BEB%7D%7BBC%7D%24%20%26%2010.%20Triangle%20proportionality%20theorem%20%5C%5C%0A11.%20%24%5Cfrac%7BAD%7D%7BDC%7D%3D%5Cfrac%7BAB%7D%7BBC%7D%24%20%26%2011.%20Substitution%20Property%20of%20equality%20%5C%5C%5B1ex%5D%20%0A%5Cend%7Btabular%7D%0A)
The valid conclusions for the manager based on the considered test is given by: Option
<h3>When do we perform one sample z-test?</h3>One sample z-test is performed if the sample size is large enough (n > 30) and we want to know if the sample comes from the specific population.
For this case, we're specified that:
- Population mean =
= $150
- Population standard deviation =
= $30.20
- Sample mean =
= $160
- Sample size = n = 40 > 30
- Level of significance =
= 2.5% = 0.025
- We want to determine if the average customer spends more in his store than the national average.
Forming hypotheses:
- Null Hypothesis: Nullifies what we're trying to determine. Assumes that the average customer doesn't spend more in the store than the national average. Symbolically, we get:
- Alternate hypothesis: Assumes that customer spends more in his store than the national average. Symbolically
where is the hypothesized population mean of the money his customer spends in his store.
The z-test statistic we get is:
The test is single tailed, (right tailed).
The critical value of z at level of significance 0.025 is 1.96
Since we've got 2.904 > 1.96, so we reject the null hypothesis.
(as for right tailed test, we reject null hypothesis if the test statistic is > critical value).
Thus, we accept the alternate hypothesis that customer spends more in his store than the national average.
Learn more about one-sample z-test here: