In: Statistics and Probability
1.) 25 Pittsburgh residents were asked how much sleep they get per night. The mean was 7.2 hours and the standard deviation was 0.78 hours. Construct and interpret a 90% confidence interval for the true mean number of hours of sleep that Pittsburgh residents get per night.
2.) A sample of 11 Chevy Volt drivers resulted in them having a mean of 42.6 miles per gallon (mpg) and a standard deviation of 4.8 mpg. Construct and interpret a 95% confidence interval for the mean gas mileage of all Chevy Volt drivers.
3.) A researcher believes the mean price of regular gas in Pennsylvania has increased from last month's mean of $2.88. A sample of 50 gas stations yields a mean price of $2.98 and a standard deviation of $0.50. Conduct a hypothesis test at the 10% significance level. Make sure to give the hypotheses, test statistic, p-value, decision, and a practical interpretation.
1)
sample mean, xbar = 7.2
sample standard deviation, s = 0.78
sample size, n = 25
degrees of freedom, df = n - 1 = 24
Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, tc = t(α/2, df) = 1.711
ME = tc * s/sqrt(n)
ME = 1.711 * 0.78/sqrt(25)
ME = 0.267
CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (7.2 - 1.711 * 0.78/sqrt(25) , 7.2 + 1.711 *
0.78/sqrt(25))
CI = (6.9331 , 7.4669)
b)
sample mean, xbar = 42.6
sample standard deviation, s = 4.8
sample size, n = 11
degrees of freedom, df = n - 1 = 10
Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.228
ME = tc * s/sqrt(n)
ME = 2.228 * 4.8/sqrt(11)
ME = 3.224
CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (42.6 - 2.228 * 4.8/sqrt(11) , 42.6 + 2.228 *
4.8/sqrt(11))
CI = (39.3755 , 45.8245)
3)
Below are the null and alternative Hypothesis,
Null Hypothesis: μ = 2.88
Alternative Hypothesis: μ > 2.88
Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (2.98 - 2.88)/(0.5/sqrt(50))
t = 1.414
P-value Approach
P-value = 0.0818
As P-value < 0.1, reject the null hypothesis.
Conclusion
There is sufficient evidence to conclude that the mean price of
regular gas in Pennsylvania has increased from last month's mean of
$2.88.