A friend of yours claims that it doesn’t really matter whether we use a normal d
ID: 3208499 • Letter: A
Question
A friend of yours claims that it doesn’t really matter whether we use a normal distribution or tdistribution in most cases since each gives you roughly the same answer. He cites the following example.
Studies show that the mean number of times students have cheated on math tests in high school is 5.6. In a random sample of 50 students, it was found that the mean number was 6.2 with a standard deviation of 2.3. Find the probability that a sample of this size will produce a mean of 6.2 or more if the actual mean is 5.6.
a) Solve using a normal distribution.
b) Solve using a t-distribution.
c) Explain to your friend why the answers are so close.
d) Explain why the t-distribution should be used in this case.
Explanation / Answer
a) here std error =std deviation/(n)1/2=2.3/(50)1/2=0.3253
hence from normal distribution P(X>6.2)=1-P(X<6.2)=1-P(Z<(6.2-5.6)/0.3253)=1-P(Z<1.8446)=1-0.9675=0.0325
b) from t distribution:; degree of freedom =50-1 =49
P(X>6.2)=P(t<1.8446)=0.0356
c) as number of observations are higher, t distribution is closer to normal distribution.
d) as we do not know population std deviation it is better to use t distribution to compensate for unknown variability.