You would like to determine if more than 40% of the observations in a population

Question

You would like to determine if more than 40% of the observations in a population are below 10. At ? = 0.05, conduct the test on the basis of the following 20 sample observations: Use Table 1. 9 7 9 9 14 8 6 12 7 13 12 8 10 13 9 10 13 5 7 7 Picture Click here for the Excel Data File a. Select the null and the alternative hypotheses. H0: p = 0.40; HA: p ? 0.40 H0: p ? 0.40; HA: p > 0.40 H0: p ? 0.40; HA: p < 0.40 b. Calculate the sample proportion. (Round your answer to 2 decimal places.) Sample proportion c. Calculate the value of test statistic. (Round your answer to 2 decimal places.) Test statistic d. The p-value is: (Round p-value to 4 decimal places.) p-value Picture 0.10 p-value < 0.01 0.01 Picture p-value < 0.025 0.025 Picture p-value < 0.05 0.05 Picture p-value < 0.10 e. What is the conclusion? Do not reject H0 since the p-value is greater than ?. Do not reject H0 since the p-value is smaller than ?. Reject H0 since the p-value is greater than ?. Reject H0 since the p-value is smaller than ?.

Explanation / Answer

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P < 0.40
Alternative hypothesis: P > 0.40

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected only if the sample proportion is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (?) and compute the z-score test statistic (z).

? = sqrt[ P * ( 1 - P ) / n ]

? = 0.10954

p = 12/20

p = 0.60
z = (p - P) / ?

z = 1.83

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is more than 1.83.

Thus, the P-value = 0.0336

Interpret results. Since the P-value (0.0336) is less than the significance level (0.05), we cannot accept the null hypothesis.

Reject H0, since the p-value is smaller than ?.

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