A random sample of 10 students contains the following...

A random sample of 10 students contains the following observations, in hours, for time spent studying in the week before final exams: $\begin{array}{llllllllll}28 & 57 & 42 & 35 & 61 & 39 & 55 & 46 & 49 & 38\end{array}$ Assume that the population distribution is normal. a. Find the sample mean and standard deviation. b. Test, at the $5 \%$ significance level, the null hypothesis that the population mean is 40 hours against the alternative that it is higher.

A random sample of 10 students contains the following observations, in hours, for time spent studying in the week before final exams:
$\begin{array}{llllllllll}28 & 57 & 42 & 35 & 61 & 39 & 55 & 46 & 49 & 38\end{array}$
Assume that the population distribution is normal.
a. Find the sample mean and standard deviation.
b. Test, at the $5 \%$ significance level, the null hypothesis that the population mean is 40 hours against the alternative that it is higher.

Key Concepts

Hypothesis Testing

Hypothesis testing is a systematic method for making decisions or inferences about population parameters based on sample data. It involves formulating a null hypothesis, which represents a statement of no effect or no difference, and an alternative hypothesis, which represents what the researcher wants to conclude. The test then evaluates whether the observed sample evidence is strong enough to reject the null hypothesis in favor of the alternative at a predetermined significance level.

One-Sample t-Test

The one-sample t-test is a statistical procedure used to determine if the mean of a sample differs significantly from a known or hypothesized population mean. It is particularly useful when dealing with small sample sizes and when the population standard deviation is unknown, relying instead on the sample standard deviation to estimate variability.

Significance Level

The significance level, often denoted by alpha (?), is the probability threshold used to determine when to reject the null hypothesis. It represents the maximum acceptable risk of a Type I error, which is the error of rejecting a true null hypothesis. A commonly used significance level in many tests is 5%, meaning there is a 5% chance of incorrectly rejecting the null hypothesis.

Sample Mean

The sample mean is a measure of central tendency that is computed by adding all the observations together and dividing by the number of observations. It provides an estimate of the average value in the sample and is used as an unbiased estimator of the population mean when the sample is random.

Sample Standard Deviation

The sample standard deviation is a measure of dispersion that quantifies how much the sample values deviate from the sample mean on average. It is calculated by taking the square root of the sample variance, which itself is computed by averaging the squared differences between each observation and the sample mean, adjusted for degrees of freedom.

Please give Ace some feedback