Question

The reading speed of second grade students in a large city is approximately​ normal, with a mean of 90 words per minute​ (wpm) and a standard deviation of 10 wpm. Complete parts​ (a) through​ (f).

​(a) What is the probability a randomly selected student in the city will read more than 95 words per​ minute?

​(b) What is the probability that a random sample of 13 second grade students from the city results in a mean reading rate of more than 95 words per​ minute? then interpret this probability.

(c) What is the probability that a random sample of 26 second grade students from the city results in a mean reading rate of more than 95 words per​ minute? then interpret.

(d) What effect does increasing the sample size have on the​ probability? Provide an explanation for this result.

(e) A teacher instituted a new reading program at school. After 10 weeks in the​ program, it was found that the mean reading speed of a random sample of 20 second grade students was 92.3 wpm. What might you conclude based on this​ result? Select the correct choice below and fill in the answer boxes within your choice.

(f) There is a​ 5% chance that the mean reading speed of a random sample of 24 second grade students will exceed what​ value?

Answer 1

Answer a)

The probability a randomly selected student in the city will read more than 95 words per​ minute is 0.3085

Answer b)

For sampling distribution of sample mean:

μ_x̄ = μ = 90

σ_x̄ = σ/sqrt(n) = 10/sqrt(13) = 2.7735

The probability that a random sample of 13 second grade students from the city results in a mean reading rate of more than 95 words per​ minute is 0.0359. This probability shows that if the mean of all the random samples of size 13 are taken from the population, then probability that this mean will be greater than 95 words per​ minute is 0.0359.

Answer c)

For sampling distribution of sample mean:

μ_x̄ = μ = 90

σ_x̄ = σ/sqrt(n) = 10/sqrt(13) = 1.9612

The probability that a random sample of 26 second grade students from the city results in a mean reading rate of more than 95 words per​ minute is 0.0054. This probability shows that if the mean of all the random samples of size 26 are taken from the population, then probability that this mean will be greater than 95 words per​ minute is 0.0054.

Answer d)

The result shows that increasing the sample size decreases the probability. This is because as the sample size increases standard error decreases. In other words, with increase in sample size, sample mean will get closer to population mean that is 90, so probability of it will be greater than 95 decreases.

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