4.6
(a)
Sampling distribution
(b)
The sample size of 15 is relatively low, so I would not necessarily expect a normal distribution and therefore, not expect a symmetric distribution. I’m not sure how to determine whether it would be right skewed or left skewed based on the data provided.
(c)
The variability of a sampling distribution is measured by the standard error. I was not enitrely sure how to calculate standard error given the data provided. I don’t know the probability that a student will say yes. I have one sample that tells me the probability is 14/15. I would have thought that the probability would be 1/2 (i.e. there is an equal chance that I student would say yes or now). If the probability is 14/15 then the standard error is 0.064. If the probability is 1/2 then the standard error is 0.129.
(d)
There will be less variability because the sample size has increased from 15 to 25.
4.12
(a)
We are 95% confident that US residents rate their mental health as not good an average of 3.40 to 4.24 days out of 30 days.
(b)
95% confident means that we are 95% certain that the range we calculated includes the mean number of days that US residents rate their mental health as not good (out of 30 days).
(c)
The interval for a 99% confidence interval will be wider.
(d)
The standard error would be smaller because as the sample size decreases, the standard error increases.
4.18
(a)
The null hypothesis is that the average number of calories consumed remained the same. The alternative hypothesis is that the average number of calories consumed is different.
H0: Average calories consumed = 1,100
HA: Average calories consumed = 1,100
(b)
The null hypothesis is that the average verbal reasoning score remained the same. The alternative hypothesis is that the average verbal reasoning score is greater than 462.
H0: Average verbal reasoning score = 462
HA: Average verbal reasoning score > 462
4.24
(a)
Yes, the observations are independent (random sample), the sample size is at least 30 (n = 36), and the distribution doesn’t have a clear skew.
(b)
HO: the average age at which gifted children first count to 10 successfully = 32 months
HA: the average age at which gifted children first count to 10 successfully < 32 months
The p-value is less than the signicance level, so I would reject the null hypothesis.
(c)
If the null hypothesis is true, the probability of observing a sample mean at least as small as 30.69 months for a sample of 36 students is only 0.034.
(d)
The 90% confidence interval is:
30.69+qnorm(0.05)*4.31/sqrt(36)
## [1] 29.50845
30.69+qnorm(0.95)*4.31/sqrt(36)
## [1] 31.87155
(e)
Yes, the null value of 32 months is not in the confidence interval so 32 months as a true average age of first counting to 10 is implausible and we would reject the null hypothesis in this scenario as well.
4.30
(a)
HO: the restaurant is not in gross violation
HA: the restaurant is in gross violation
(b)
Type 1 error is find the restaurant to be in gross violation of the rules when it actually is not in violation.
(c)
Type 2 error is find the restaurant to be not in gross violation of the rules when it actually is in violation.
(d)
Type 1 error is more problematic for the restaurant owner because his/her license to serve food will be revoked when it shouldn’t be.
(e)
Type 2 error is more problematic for diners because the restaurant will be in violation of safety regulations and will be allowed to continue serving food.
(f)
I would prefer the inspector requires strong evidence because I would want to minimze Type 2 errors.
4.32
(a)
True because 99% confidence interval just expands a 95% confidence interval further.
(b)
False because decreasing the significance level reduces the probability of making a Type 1 error. The correct statement would be Decreasing the significance level will decrease the probability of making a Type 1 Error.
(c)
False because even though we are failing to reject that the mean is 5, we are not saying it is 5. We could correct the statement by saying the true population mean might be 5.
(d)
True because the power of a test is 1 minus the probability of making a Type 2 error.
(e)
True because larger samples create more precise point estimates.
4.38
Plot A is (3) a distribution of 500 sample means from random samples of each size 81 it is more precise than Plot C given it has a larger sample size.
Plot B is (1) a single random sample of 500 values from this population because it most closely mirrors the population distribution.
Plot C is (2) a distribution of 500 sample means from random samples of each size 18 because its data is all relatively close to the mean but it is not as precise as Plot A given it has a smaller sample size.
4.44
(a)
H0: Average % of nearsighted children = 8%
HA: Average % of nearsighted children = 8%
(b)
21/194
## [1] 0.1082474
(c)
((21/194)-0.08)/0.0195
## [1] 1.448586
(d)
1-pnorm((21/194),0.08,0.0195)
## [1] 0.07372665
(e)
The conculsion of the hypothesis test depends on the significance level. If the significance level is greater than 0.074 then we would reject the null hypothesis.