Question: 6. If If ? = ?? ?/? has a t distribution with n-1 degrees of freedom, show that a (1)100% = ?? ?/? has a
6. If If ? =
??
?/?
has a t distribution with n-1 degrees of freedom, show that a (1)100% =
??
?/?
has a t distribution with n-1 degrees of freedom, show that a (1)100%
confidence interval for ? is given by? ?
2
,?1
?
?
.
Section C: Confidence interval for two samples
7. The following data table gives the daily amount of milk produced by each cow in two herds
(in L). Suppose it is assumed that the standard deviation of milk produced by a cow is equal
across the two herds, and that the daily amount of milk produced by a cow is normally
distributed. Give a 99% confidence interval for the difference between the mean amount of
milk produced per cow in herd 1 and the mean amount of milk produced per cow in herd
2.
Herd 1 12.4 12.7 11.8 11.6 10.3 9.4 11.7 12.1
Herd 2 11.3 10.5 9.6 10.2 10.0 11.1 11.4
8. In a study of the rate at which the pesticide hexaconazole is absorbed through skin, skin
specimens were exposed to 24 g of hexaconazole. Four specimens were exposed for 30
minutes and four others were exposed for 60 minutes. The amounts (in g) that were
absorbed were:
30 minutes: 3.1 3.3 3.4 3.0
60 minutes: 3.7 3.6 3.7 3.4
It is assumed that the data are normally distributed. It is also assumed that the absorption
over 30 minute exposure has a standard deviation of 0.1 while exposure over 60 minutes
has a standard deviation of 0.12. Give a 95% confidence interval for the mean amount
absorbed in the time interval between 30 and 60 minutes after exposure.
9. A researcher is studying the relationship between sugar consumption and adult onset of
diabetes. She takes a random sample of 40 adults with high sugar consumption and 44
adults with low sugar consumption. In the first sample, 17 people have diabetes while inthe second sample, 11 people have diabetes.
a) Give a 90% confidence interval for the difference between the proportions of people
with high sugar consumption who have diabetes and the proportion of people with low
sugar consumption who have diabetes.
b) What conclusions could this researcher draw about the relationship between sugar
consumption and adult onset of diabetes? Explain.
10. In a random sample of 340 cars driven at low altitudes, 46 of them produced more than 10
g of particulate pollution per L of fuel consumed. In a random sample of 85 cars driven at
high altitudes, 21 of them produced more than 10 g of particulate pollution per L of fuel
consumed. Find a 98% confidence interval for the difference between the proportions for high-altitude and low-altitude vehicles.
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