5. Tickets to a junior high school play
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Answer:
We conclude that the population proportion is equal to 0.70.
Step-by-step explanation:
We are given that a random sample of 100 observations from a binomial population gives a value of pˆ = 0.63 and you wish to test the null hypothesis that the population parameter p is equal to 0.70 against the alternative hypothesis that p is less than 0.70.
Let p = <u><em>population proportion.</em></u>
(1) The intuition tells us that the population parameter p may be less than 0.70 as the sample proportion comes out to be less than 0.70 and also the sample is large enough.
(2) So, Null Hypothesis, : p = 0.70 {means that the population proportion is equal to 0.70}
Alternate Hypothesis, : p < 0.70 {means that the population proportion is less than 0.70}
The test statistics that would be used here <u>One-sample z-test</u> for proportions;
T.S. = ~ N(0,1)
where, = sample proportion = 0.63
n = sample of observations = 100
So, <u><em>the test statistics</em></u> =
= -1.528
The value of z-test statistics is -1.528.
<u>Now at 0.05 level of significance, the z table gives a critical value of -1.645 for the left-tailed test.</u>
Since our test statistics is more than the critical value of z as -1.528 > -1.645, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which <u><em>we fail to reject our null hypothesis</em></u>.
Therefore, we conclude that the population proportion is equal to 0.70.
(c) The observed level of significance in part B is 0.05 on the basis of which we find our critical value of z.