Question

we want to conduct a hypothesis test of the claim that the population mean reading speed of second graders is different from 29.4 words per minute. so, we choose a random sample of students' reading speeds. the sample has a mean of 29.5 words per minute and a standard deviation of 3.8 words per minute. for each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean. then calculate that statistic. round your answers to two decimal places. (a) the sample has size 15, and it is from a population with a distribution about which we know very little. 0

Answer 1

In a scenario with a sample size of 15 and limited information about the population distribution, the t-test statistic for the hypothesis test on the population mean reading speed
`(\(\bar{x} = 29.5\), \(s = 3.8\), \(n = 15\), \(\mu_0 = 29.4\))` is approximately
`\(0.10\)`.

In this scenario, we're testing the hypothesis about the population mean reading speed of second graders against a claimed value of 29.4 words per minute.

Given:

Sample mean
`(\(\bar{x}\))` = 29.5 words per minute

Sample standard deviation
`(\(s\))` = 3.8 words per minute

Sample size
`(\(n\))` = 15

Claimed population mean
`(\(\mu_0\))` = 29.4 words per minute

Since the population distribution is unknown and the sample size is less than 30, we'll use the t-distribution and the t-test statistic.

The formula for the t-test statistic for a one-sample t-test is:

`\[ t = \frac{\bar{x} - \mu_0}{(s)/(√(n))} \]`

Where:

`\(\bar{x}\)` = Sample mean

`\(\mu_0\)` = Claimed population mean

`\(s\)` = Sample standard deviation

`\(n\)` = Sample size

Plugging in the given values:

`\[ t = (29.5 - 29.4)/((3.8)/(√(15))) \]`

Calculating:

`\[ t = (0.1)/((3.8)/(√(15))) \]`

`\[ t \approx (0.1)/(0.9822) \]`

`\[ t \approx 0.10 \]`

Therefore, the t-test statistic for this sampling scenario is approximately
`\(0.10\)` (rounded to two decimal places).