Statistics P and T Values Formulas
Essay by buttercup6117 • July 8, 2012 • Study Guide • 433 Words (2 Pages) • 1,669 Views
A candidate must gather at least 8,000 valid signatures on a petition before the deadline in order to run in an election. One candidate turns in 10,000 signatures right before the deadline, but its always expected that some percentage of them are invalid. Election officials take a random sample of 100 signatures and thoroughly investigate them to find that 84% are valid. Is this statistically significant evidence that the candidate has enough valid signatures overall? Explain.
The population of interest is the 10,000 signatures turned in by the candidate. Let be the unknown population proportion. The candidate has enough valid signatures if the population proportion is or greater.
The estimate of is the sample proportion , the proportion of valid signatures in the sample of 100.
0.840 (Where 84, and 100)
The null and alternative hypotheses:
0.80
0.80
The test statistic:
1.000 ( 0.840, 0.80, and 100)
If the null hypothesis is true, the observed value of the test statistic, 1.000, is a randomly selected observation from a standard normal distribution.
The rejection region:
The critical value of the rejection region is1.645 (this is the 95th percentile of the standard normal distribution). This is the z-value that cuts off an area of = 0.05 in the right tail of a standard normal distribution.
Decision rule: reject the null hypothesis and conclude the alternative is true if the test statistic is greater than the critical value:
Reject if 1.645
The p-value
The p-value is the probability of observing a test statistic as extreme as or more extreme than the one we observed (assuming the null hypothesis is true).
Expressed another way, it's the probability that a randomly selected observation from a standard normal distribution would be "as extreme as or more extreme than" the value of the test statistic.
P-value = 1.000) = 0.1587
Decision rule: reject the null hypothesis and conclude the alternative is true if the p-value is less than the level of significance.
Reject if: p-value < 0.05
Conclusion
The test statistic does not fall in the rejection region. At the 0.05 level of significance, there is insufficient evidence to reject the null hypothesis.
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