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11 June, 00:31

400 students were randomly sampled from a large university, and 289 said they did not get enough sleep. Conduct a hypothesis test to check whether this represents a statistically significant difference from 50%, and use a significance level of 0.01.

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  1. 11 June, 04:05
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    The sample proportion represents a statistically significant difference from 50%

    Step-by-step explanation:

    Null hypothesis: The sample proportion is the same as 50%

    Alternate hypothesis: The sample proportion is not the same as 50%

    z = (p' - p) : sqrt[p (1 - p) : n]

    p' is sample proportion = 289/400 = 0.7225

    p is population proportion = 50% = 0.5

    n is number of students sampled = 400

    z = (0.7225 - 0.5) : sqrt[0.5 (1 - 0.5) : 400] = 0.2225 : 0.025 = 8.9

    The test is a two-tailed test. Using a 0.01 significance level, critical value is 2.576. The region of no rejection of the null hypothesis is - 2.576 and 2.576.

    Conclusion:

    Reject the null hypothesis because the test statistic 8.9 falls outside the region bounded by the critical values - 2.576 and 2.576.

    There is sufficient evidence to conclude that the sample proportion represents a statistically significant difference from 50%.
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