An automobile manufacturer claims that a particular model gets 28 miles to the gallon. The
Environmental Pollution Agency, using a sample of 49 automobiles of this model, finds the sample
mean to be 26.8 miles per gallon. From previous studies, the population standard deviation is known
to be 5 miles per gallon. Could we reasonably expect (within 2 standard errors) that we could select
such a sample if indeed the population mean is actually 28 miles per gallon?
Bbn
To determine if we could reasonably expect the sample mean of 26.8 miles per gallon, based on the population mean of 28 miles per gallon, we can perform a hypothesis test using the z-test.
First, let's define the null and alternative hypothesis:
Null hypothesis (H0): The population mean is 28 miles per gallon.
Alternative hypothesis (Ha): The population mean is not 28 miles per gallon.
We will use a significance level of 0.05 (5%).
Next, let's calculate the standard error of the sample mean:
Standard Error = Population Standard Deviation / Square Root of Sample Size
Standard Error = 5 / √49 = 5 / 7 = 0.714 miles per gallon
Now, let's calculate the z-score:
z = (Sample Mean - Population Mean) / Standard Error
z = (26.8 - 28) / 0.714 = -1.889
We can now find the critical value(s) for a two-tailed test with a significance level of 0.05. The critical value for a 95% confidence level is ±1.96.
Since our calculated z-score of -1.889 falls within the range of -1.96 to +1.96, we fail to reject the null hypothesis. This means that we have insufficient evidence to conclude that the population mean is significantly different from 28 miles per gallon.
Therefore, based on the given information and using a 5% significance level, we can reasonably expect to select such a sample even if the population mean is actually 28 miles per gallon.
To determine whether we can reasonably expect the sample mean of 26.8 miles per gallon to be within 2 standard errors of the population mean of 28 miles per gallon, we need to calculate the margin of error and compare it to the range.
The margin of error is calculated using the formula:
Margin of Error = Critical Value * Standard Error
The critical value is determined based on the desired level of confidence. For this analysis, let's assume a 95% confidence level, which corresponds to a critical value of 1.96.
The standard error is calculated using the formula:
Standard Error = Population Standard Deviation / Square Root of Sample Size
Given that the population standard deviation is known to be 5 miles per gallon and the sample size is 49, we can plug in the values:
Standard Error = 5 / Square Root of 49
= 5 / 7
≈ 0.7143
Next, we can calculate the margin of error:
Margin of Error = 1.96 * 0.7143
≈ 1.400
To determine whether the sample mean of 26.8 miles per gallon is within 2 standard errors of the population mean of 28 miles per gallon, we need to calculate the range:
Range = (Population Mean - Margin of Error, Population Mean + Margin of Error)
Range = (28 - 1.400, 28 + 1.400)
= (26.6, 29.4)
Since the sample mean of 26.8 miles per gallon falls within the range (26.6, 29.4), we can reasonably expect, within 2 standard errors, that we could select such a sample if the population mean is actually 28 miles per gallon.