Question

it is believed that lake tahoe community college (ltcc) intermediate algebra students get less than 7 hours of sleep per night, on average. a survey of 22 ltcc intermediate algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. at a level of significance of 5%, do ltcc intermediate algebra students get less than 7 hours of sleep per night, on average? the type ii error is not to reject that the mean number of hours of sleep ltcc students get per night is at least seven when, in fact, the mean number of hours is

Answer 1

The Type II error is not rejecting the null hypothesis when, in fact, the mean number of hours of sleep LTCC students get per night is less than seven. Thus, the correct answer is: d. is less than seven hours.

What is a type II error?

To determine whether LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average, we can conduct a hypothesis test.

Set up the hypotheses:

Null Hypothesis (H0): The mean number of hours of sleep LTCC Intermediate Algebra students get per night is equal to seven (μ = 7).

Alternative Hypothesis (H1): The mean number of hours of sleep LTCC Intermediate Algebra students get per night is less than seven (μ < 7).

We'll use a one-sample t-test since we have the sample mean, standard deviation, and a relatively small sample size (n = 22).

Given:

Sample mean (x) = 7.24

Standard deviation (σ) = 1.93

Sample size (n) = 22

Significance level (α) = 0.05 (or 5% level of significance)

Next, calculate the test statistic (t-score) using the formula:

t = (x - μ) / (σ /
`\sqrt`(n))

Substituting the given values:

t = (7.24 - 7) / (1.93 /
`\sqrt`(22))

To find the critical t-value, determine the degrees of freedom (df = n - 1 = 22 - 1 = 21) and the critical region based on the significance level.

Using a t-table or statistical software, we find that the critical t-value for a one-tailed test with a significance level of 0.05 and 21 degrees of freedom is approximately -1.721.

Now, calculate the t-score:

t = (7.24 - 7) / (1.93 /
`\sqrt`(22))

Calculating this expression:

t ≈ 0.287

Since the calculated t-value (t ≈ 0.287) is greater than the critical t-value (-1.721), we fail to reject the null hypothesis.

Therefore, at the 5% level of significance, we do not have sufficient evidence to conclude that LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average.

Now let's address the Type II error.

The Type II error is not rejecting the null hypothesis when, in fact, the mean number of hours of sleep LTCC students get per night is less than seven. In other words, the mean number of hours of sleep could be less than seven, but we failed to detect it.

Thus, the correct answer is:

d. is less than seven hours.