Question

1. select the equation that has an exponential growth of 7% a) ( f(x)=7(1.1)^{x} ) b) ( f(x)=25(1.07)^{x} ) c) ( f(x)=25(0.93)^{x} ) d) ( f(x)=7(.99)^{x} ) 10. a population characteristic is normally distributed. approximately what percentage of the population lies within three standard deviations of the mean? a) 95% b) 34% c) 99.7% d) 68%

Explanation:

Step1: Recall the exponential growth formula

The general form of an exponential growth function is \(f(x)=a(1 + r)^x\), where \(a\) is the initial value and \(r\) is the growth rate.

Step2: Analyze each option

• For option a: \(f(x)=7(1.1)^x\), the growth rate \(r = 0.1=10\%\)
• For option b: \(f(x)=25(1.07)^x\), using the formula \(1 + r=1.07\), so \(r = 0.07 = 7\%\)
• For option c: \(f(x)=25(0.93)^x\), since \(1 + r=0.93\), \(r=- 0.07\) (this is exponential decay)
• For option d: \(f(x)=7(0.99)^x\), since \(1 + r=0.99\), \(r=-0.01\) (this is exponential decay)

Step1: Recall the empirical rule for normal distribution

The empirical rule (68 - 95 - 99.7 rule) states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation of the mean (\(\mu\pm\sigma\))
• Approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean (\(\mu\pm2\sigma\))
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations of the mean (\(\mu\pm3\sigma\))

Answer:

B. \(f(x)=25(1.07)^x\)