55. It is believed that Lake Tahoe Community College (LTCC)...

55. It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with standard deviation of 1.93 hours. At a level of significance of $5 \%,$ do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average? The distribution to be used for this test is $\bar{X} \sim$ _________ a. $\quad N\left(7.24, \frac{1.93}{\sqrt{22}}\right)$ b. $\quad N(7.24,1.93)$ c. $t_{22}$ d. $t_{21}$

55. It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with standard deviation of 1.93 hours. At a level of significance of $5 \%,$ do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average? The distribution to be used for this test is $\bar{X} \sim$ _________
a. $\quad N\left(7.24, \frac{1.93}{\sqrt{22}}\right)$
b. $\quad N(7.24,1.93)$
c. $t_{22}$
d. $t_{21}$

Key Concepts

Degrees of Freedom

Degrees of freedom are a parameter used in many statistical distributions, including the t-distribution, and represent the number of independent values that are free to vary when estimating a parameter. For a sample size of n, the degrees of freedom for the t-distribution are determined as n - 1. This adjustment is important to accurately reflect the variability in estimates when the sample standard deviation is used in place of the unknown population standard deviation.

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of the sample mean, which is obtained by taking all possible samples of a given size from a population. When the population standard deviation is known, the sample mean is normally distributed; however, when the population standard deviation is unknown and estimated from the sample, the sampling distribution is better approximated by the t-distribution, especially for small sample sizes.

Student's t-Distribution

The t-distribution is used when the sample size is small and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the extra uncertainty from estimating the standard deviation from the sample. The t-distribution is characterized by degrees of freedom, which depend on the sample size, and in this context, it is applied to construct confidence intervals or perform hypothesis tests on the sample mean.

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