Several drugs are used to treat diabetes. A sales specialist...

Several drugs are used to treat diabetes. A sales specialist for a leading pharmaceutical company needs an estimate of the number of new prescriptions that were written during a particular month for his company's new diabetes drug. The numbers of new prescriptions in a random sample of 25 sales districts are as follows: $\begin{array}{llllllllll}210 & 240 & 190 & 275 & 290 & 185 & 223 & 190 & 185 & 192 \\ 265 & 312 & 284 & 261 & 243 & 168 & 240 & 170 & 187 & 190 \\ 215 & 240 & 210 & 235 & 290 & & & & & \end{array}$ a. Find a $90 \%$ confidence interval for the average number of new prescriptions written for this new drug among all the sales districts. State the assumptions. b. Calculate the widths for $95 \%$ and $98 \%$ confidence intervals.

Several drugs are used to treat diabetes. A sales specialist for a leading pharmaceutical company needs an estimate of the number of new prescriptions that were written during a particular month for his company's new diabetes drug. The numbers of new prescriptions in a random sample of 25 sales districts are as follows:
$\begin{array}{llllllllll}210 & 240 & 190 & 275 & 290 & 185 & 223 & 190 & 185 & 192 \\ 265 & 312 & 284 & 261 & 243 & 168 & 240 & 170 & 187 & 190 \\ 215 & 240 & 210 & 235 & 290 & & & & & \end{array}$
a. Find a $90 \%$ confidence interval for the average number of new prescriptions written for this new drug among all the sales districts. State the assumptions.
b. Calculate the widths for $95 \%$ and $98 \%$ confidence intervals.

Key Concepts

Effect of Confidence Level on Interval Width

The width of a confidence interval is directly influenced by the chosen confidence level. Higher confidence levels correspond to larger critical values, thereby increasing the margin of error and resulting in a wider interval. This reflects a trade-off between the certainty that the interval contains the true parameter and the precision of the interval estimate.

Margin of Error

The margin of error represents the range of uncertainty around the point estimate and is calculated using the critical value from the t-distribution multiplied by the standard error of the sample. It defines how far the true parameter value is expected to lie from the sample estimate at a specified confidence level.

t-distribution

The t-distribution is used instead of the normal distribution when constructing confidence intervals, particularly when the sample size is small and the population standard deviation is unknown. Its heavier tails account for additional variability, ensuring that the interval appropriately reflects the uncertainty in the estimate.

Degrees of Freedom

Degrees of freedom, defined as the sample size minus one (n-1), refer to the number of values in the data that are free to vary when estimating statistical parameters. They are crucial in determining the shape of the t-distribution used in confidence interval calculations, affecting the area under the curve and the critical values.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. It quantifies the uncertainty inherent in estimating the population parameter by providing an upper and lower limit that, with a certain level of confidence, is believed to enclose the true parameter value.

Transcript

00:01 So we have a sample of sales where n is 25, and we have no indication.

00:04 I'm assuming there would be many, many.

00:06 It doesn't say that the population size is, you know, say 50 companies and they're doing 25.

00:12 So i'm going to assume that there's a large, that n is large.

00:17 I'm also going to assume that these are approximately normally distributed.

00:22 And i'm also going to assume that these are simple random samples of different locations.

00:29 And so when i calculate the value here, and let me quick type in, we want a 90 % confidence interval in part a, a 90 % confidence interval for the mean.

00:49 And when i calculate the values, i get that the x bar here is 227 .5, excuse me, six.

00:59 And the sample standard deviation or the variability of those is 41 .86, and i'll call it 4.

01:07 Now, the t star value, we know our alpha value is 0 .1, therefore, the t value with 0 .05 in the upper tail, in the upper tail, in the upper tail, in the upper tail, in the upper tail, that is going to correspond with set 1 .1 .7, let me put the degrees of freedom first.

01:18 With the degrees of freedom of 24 and an upper tail of 0 .05, that is going to correspond with set one point, i'll put it this way, 1 .711.

01:34 So we would have our 227 .6 plus or minus that 1 .711 times the sample standard deviation of 41 .864 divided by the square root of 25 or 5.

01:51 And i believe those values come out to be 213 .28 to 241 .92.

02:03 So there is our confidence interval.

02:06 90 % confident.

02:07 Now part b asks us to find a width, actually two widths for a 95 % confidence interval for the mean and a 98 % confidence interval for the mean...

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