Problem 4 Determine the critical value for... [FREE SOLUTION] (2024)

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When we talk about the t-distribution, we're referring to a type of probability distribution that is symmetrical around the mean. Unlike the normal distribution, the t-distribution has heavier tails, which means it has a greater chance of producing values far from its mean. This makes it especially useful when dealing with small sample sizes where the population standard deviation (\( \sigma \)) is unknown.

The t-distribution is crucial in hypothesis testing, particularly for the t-test. It's used because it accounts for the variability introduced by estimating the population standard deviation from the sample. As sample size increases, the t-distribution approaches the shape of the normal distribution.

Key points about the t-distribution:

• Symmetrical and bell-shaped, like the normal distribution
• Heavier tails than the normal distribution, providing a higher probability for extreme values
• Defined by degrees of freedom which influence its shape
• Used primarily in small sample sizes when the population standard deviation is unknown.

Degrees of freedom (df) are a concept used in statistical analyses to describe the number of independent values that can vary in an analysis without breaking any constraints. In the context of the t-distribution, degrees of freedom are calculated as the sample size minus one ( ).

Degrees of freedom are crucial because they affect the shape of the t-distribution. With fewer degrees of freedom, the t-distribution is wider and has heavier tails. As the degrees of freedom increase, the t-distribution narrows and resembles a normal distribution more closely.

Practical points about degrees of freedom:

• It essentially boils down to the number of datapoints that are free to vary
• In smaller samples, fewer degrees of freedom result in a broader distribution
• Understanding degrees of freedom helps in choosing the correct t-distribution for hypothesis testing

The significance level () is the probability of rejecting the null hypothesis when it is true. It is a threshold set by the researcher to determine when there is enough evidence to conclude that an effect or difference exists.

A common significance level is 0.05, which means there's a 5% risk of concluding that a difference exists when there actually is none. In a left-tailed test, this value marks the boundary below which we would reject the null hypothesis in favor of the alternative hypothesis.

Key aspects of the significance level:

• It is denoted by alpha ( )
• It defines the rejection region for the hypothesis test
• Common values are 0.01, 0.05, and 0.10, with 0.05 being particularly standard
• In a left-tailed test, we look at the lower tail; for a right-tailed or two-tailed test, it differs accordingly