QNT week 5

This one you can use for some parts

One-Sample Hypothesis Testing Cases

One-Sample Hypothesis Testing Cases

A sample of 765 voters were taken at the exit poll who were asked for whom they voted. During this time the two presidential candidates were Republican George Bush and Democratic Al Gore. Out of the sample of 765 voters, 358 elected Al Gore and 407 elected George Bush. After collecting this data, predicting the significant level at 0.10 may decide that George Bush will receive more than 50% votes. It may be possible that a candidate could win if he gets more than 50% votes. To determine that one will apply the test of proportion to testing the hypothesis.

The alternate and null hypothesis can be determined as :

Alternate Hypothesis: H1: p > 0.50 and the Null Hypothesis: H0: p = 0.50, defining it as the right tail test. The sample size larger than 30, one can reference the Z-test to test the hypothesis. Significance level at α = 0.10. Critical value at the same level of 0.10 is calculated as 1.28. Greater than critical value 1.28 is the Critical region area. Decision Rule: it the test results falls in the critical region, the null hypothesis may be rejected (Black, 2017). For instance, null hypothesis is rejected if: Z test statistic > Critical value Z0.10 = 1.28. Test statistic is calculated as: z = , Where, p0 = null hypothesis proportion = 0.5, x = number of voters who voted for George W. Bush = 407 n = 765. Therefore, Z test statistic = = 1.7716. Resulting the Z-test statistic > critical value, in other words the test falls in the rejection region. Concluding the null hypothesis is rejected at 0.10 significance level which is shown in the graph below. The right of the z value 1.282 is the rejection region which is in red. 1.772 is the test statistic which falls in the rejection area determining the null hypothesis is rejected. Resulting George Bush will get more than 50% votes by accepting the alternate hypothesis.

Conclusion

Using the null and alternate hypothesis to calculate if the Republicans or Democratic will receive more votes, it is evident by what was configured above Republican George Bush may be win the election. This may be telecast across the world at 8:01 PM that George W. Bush will win selective states.

Case Study – SpeedX

Presently, mean and standard deviation of the total time taken by customers of SpeedX to pay bills are 24 days and 6 days correspondingly. Including self-addressed envelope and stamped with invoices is anticipated to decrease payment period by 2 days. To see if sending stamped and self-addressed envelope with invoices decreases payment period from 24 days to 22 days, a pilot study was conducted. In this pilot study, we choose 220 customers as a sample, and self-addressed stamped envelopes were sent to the customers. To determine the number of days it would take for the consumers to pay their bills was documented. Per this data, hypothesis test is to be showed at 0.10 significance level to check if payment period has decreased to 22 days.

The null hypothesis and alternate hypothesis are stated as: Null Hypothesis H0: µ = 22 and Alternate hypothesis H1: µ < 22. This is a left-tailed test for single population mean. It means, the rejection area will be left to the critical value (Black, 2017). Decision rule: Null hypothesis is rejected if test statistic is less than critical value. As sample size is greater than 30, we can use z-test for hypothesis testing. At α = 0.10, critical z-value is -1.28. Thus, null value is rejected if: z test statistic < Critical value, for example z test statistic < -1.28

The test statistic is calculated as:

Z test statistic =

Where,

µ0 = 22

σ = 6

n = 220

Sample mean = 21.63181818 (calculated from given data)

Therefore,

Z test statistic = -0.9102

Critical value = -1.28

Furthermore, test statistic > critical value.

Therefore, the test statistic does not fall in the rejection region. This displayed on the graph below. In this graph, the rejection region is the area left to the critical value -1.282. The rejection region is shaded in red. The calculated test statistic does not fall under the shaded region. Thus, we fail to reject null hypothesis at significance level of 0.10.

Conclusion

There is not enough evidence to show that sending stamped and self-addressed envelopes to customers reduces payment time by 2 days.

References

Black, K. (2017). Business statistics for contemporary decision making (9th ed.). Hoboken, NJ: John Wiley & Sons, Inc.

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Dashed line is the test statistic and shaded are is the critical region

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-1.282

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-0.910

Critical Region and the test statistic