Mary conducted a study to investigate whether first year science students were more likely than University students in general to use laptops (or tablets) to take notes in lectures. Within the population of university students only 20% reported that they used laptops or tablets to take notes in lectures. A random sample of 40 first year science students was selected and it was found that 14 students reported using laptops or tablets to take notes in lectures.Mary assumed first year science students were no different from University students in general with regards to their use of laptops (tablets) in lectures. She then used a normal approximation to determine the probability of observing 14 or more students, out of 40, using laptops (tablets). Which one of the following statements is false? Group of answer choicesIf she didn't use a continuity correction her probability will be an underestimate of the exact probabilityIf she did use a continuity correction her probability will still be an underestimate of the exact probabilityAssuming Mary's model is correct, the exact probability of observing 14 students using a laptop or tablet to take notes in lectures is 0.0115If Mary repeated this study many times then approxiately 14 times out of 1000 she should expect to observe 14 or more first year science students using a laptop or tablet (to take notes in lectures).Based on evidence form Mary's sample, first year science students seem to be more likley to use laptops (tablets) in lectures than the university students in general.
Question
Mary conducted a study to investigate whether first year science students were more likely than University students in general to use laptops (or tablets) to take notes in lectures. Within the population of university students only 20% reported that they used laptops or tablets to take notes in lectures. A random sample of 40 first year science students was selected and it was found that 14 students reported using laptops or tablets to take notes in lectures.Mary assumed first year science students were no different from University students in general with regards to their use of laptops (tablets) in lectures. She then used a normal approximation to determine the probability of observing 14 or more students, out of 40, using laptops (tablets). Which one of the following statements is false? Group of answer choicesIf she didn't use a continuity correction her probability will be an underestimate of the exact probabilityIf she did use a continuity correction her probability will still be an underestimate of the exact probabilityAssuming Mary's model is correct, the exact probability of observing 14 students using a laptop or tablet to take notes in lectures is 0.0115If Mary repeated this study many times then approxiately 14 times out of 1000 she should expect to observe 14 or more first year science students using a laptop or tablet (to take notes in lectures).Based on evidence form Mary's sample, first year science students seem to be more likley to use laptops (tablets) in lectures than the university students in general.
Solution
The problem involves a binomial distribution with n = 40 trials and a success probability of p = 0.20. We are interested in the probability of observing k = 14 or more successes.
The mean of a binomial distribution is μ = np = 40 * 0.20 = 8, and the standard deviation is σ = sqrt(np(1-p)) = sqrt(40 * 0.20 * 0.80) = 4.
We can use a normal approximation to the binomial distribution to estimate the probability of observing 14 or more successes. The z-score for k = 14 is z = (14 - 8) / 4 = 1.5.
Looking up 1.5 in a standard normal table, we find that the probability that a standard normal variable is less than 1.5 is approximately 0.9332. Therefore, the probability that a standard normal variable is greater than 1.5 is 1 - 0.9332 = 0.0668.
This is the probability of observing 14 or more successes without using a continuity correction. If we use a continuity correction, we would subtract 0.5 from 14 before calculating the z-score, which would give us a slightly smaller z-score and a slightly larger probability.
Therefore, the statement "If she didn't use a continuity correction her probability will be an underestimate of the exact probability" is false. Without a continuity correction, the probability will be an overestimate, not an underestimate.
The other statements are true. The exact probability of observing 14 successes is indeed 0.0115, which is much smaller than the normal approximation. If Mary repeated the study many times, she would expect to observe 14 or more successes about 14 times out of 1000. And based on the evidence from Mary's sample, first year science students do seem to be more likely to use laptops or tablets in lectures than university students in general.
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