Refer to Exercise 2.65(p .59) and the Teaching of Psychology (May 1998) study in which students

step 1 So we have a sample of size 15, and we have the mean of those values is 5 .87, and the sample st

Refer to Exercise 2.65(p .59) and the Teaching of Psychology...

Refer to Exercise $2.65(\mathrm{p} .59)$ and the Teaching of Psychology (May 1998) study in which students assisted in the training of zoo animals. A sample of 15 psychology students rated "The Training Game" as a "great" method of understanding the animal's perspective during training on a $7-1$ point scale (where $1=$ strongly disagree and $7=$ strongly agree $)$.The mean response was $5.87,$ with a standard deviation of 1.51 a. Construct a $95 \%$ confidence interval for the true mean response of the students. b. Suppose you want to reduce the width of the $95 \%$ confidence interval to half the size obtained in part a. How many students are required in the sample in order to obtain the desired confidence interval width?

Refer to Exercise $2.65(\mathrm{p} .59)$ and the Teaching of Psychology (May 1998) study in which students assisted in the training of zoo animals. A sample of 15 psychology students rated "The Training Game" as a "great" method of understanding the animal's perspective during training on a $7-1$ point scale (where $1=$ strongly disagree and $7=$ strongly agree $)$.The mean response was $5.87,$ with a standard deviation of 1.51
a. Construct a $95 \%$ confidence interval for the true mean response of the students.
b. Suppose you want to reduce the width of the $95 \%$ confidence interval to half the size obtained in part a. How many students are required in the sample in order to obtain the desired confidence interval width?

Key Concepts

Sample Size Determination

Determining the required sample size to achieve a desired confidence interval width involves understanding how the sample size affects the margin of error. Larger samples yield a smaller standard error, which in turn leads to a narrower confidence interval. Sample size calculations involve rearranging the margin of error formula to solve for the number of observations needed to meet a specified precision.

Margin of Error

The margin of error represents the maximum expected difference between the sample statistic and the true population parameter. It is calculated using the product of the critical value (from the t- or z-distribution) and the standard error of the estimate, and it determines the precision of the confidence interval.

t-Distribution

The t-distribution is used instead of the normal distribution when constructing confidence intervals from small samples and the population standard deviation is unknown. It accounts for the additional uncertainty arising from estimating the standard deviation from the sample, and its shape is determined by the degrees of freedom (typically the sample size minus one).

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter with a specified level of confidence. It combines the sample estimate (such as the mean) with a margin of error, which reflects the estimated variability in the data and the chosen confidence level.

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