Question

question #7
the average number of candies per bag is estimated to be between 21 and 35. determine the sample mean.
28
30
32
60
question #8
for a population that is distributed normally with a mean of 62 and a standard deviation of 15, calculate p(68 < x < 88).
.3031
.0310
.7341
.9690

Explanation:

Step1: Recall sample - mean formula for confidence interval

The sample mean for a confidence interval is the mid - point of the interval. The formula for the mid - point of an interval \([a,b]\) is \(\bar{x}=\frac{a + b}{2}\), where \(a = 21\) and \(b = 35\).

Step2: Calculate the sample mean

\(\bar{x}=\frac{21+35}{2}=\frac{56}{2}=28\)

Step3: For normal distribution probability calculation

First, standardize the values using the formula \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 62\) and \(\sigma = 15\).
For \(x = 68\), \(z_1=\frac{68 - 62}{15}=\frac{6}{15}=0.4\)
For \(x = 88\), \(z_2=\frac{88 - 62}{15}=\frac{26}{15}\approx1.73\)

Then, use the standard normal distribution table. \(P(68<X<88)=P(0.4<Z<1.73)=P(Z < 1.73)-P(Z < 0.4)\)
From the standard - normal table, \(P(Z < 1.73)=0.9582\) and \(P(Z < 0.4)=0.6554\)
\(P(0.4<Z<1.73)=0.9582 - 0.6554=0.3028\approx0.3031\)

Answer:

Question #7: A. 28
Question #8: A. 0.3031