Question

1. You are going to write an article for the monthly publication Consumer Digest.  Radon is an invisible gas in homes that left unattended may lead to cancer.  Many homebuyers buy detectors to check the level in their homes.  How accurate are these detectors?   To answer this question, university researchers placed 33 radon detectors Model Type RSII in a chamber that that exposed them to 105 picocuries per liter of radon.  The detector readings were as follows

91.9     97.8      111.4     122.3     105.4    95.0    103.8    99.6      119.3   104.8     101.7   92.1   97.6   111.1   125.9   105.4       95.0      103.7     99.7    119.1     98.1   105.0   101.6    92.2       97.6     91.6

111.5   122.2   104.9       95.5       103.5    101.1   74.9

Please review the product and include the following:

Use all three method to analyze this sample data.

1. Check for outliers, if there are any remove them and analyze the remaining data
2. Traditional Approach (z-score or t-score & look at distance from the mean)
3. P-Value
4. Interval at 95%

Answer 1

Solution: Mean of the sample data = 99.10 , Std Dev = 13.34

Sample Data
91.9
97.8
111.4
122.3	Mean	99.10
105.4	Std. Dev	13.33877
95.0
103.8
99.6
119.3
104.8
101.7
92.1
97.6
111.1
125.9
105.4
95.0
103.7
99.7
119.1
98.1
105.0
101.6
92.2
97.6
91.60
111.5
122.20
104.90
95.50
103.50
101.10
74.90

Now, For outlier calculation, we know that

We can cover more of the data sample if we expand the range as follows:

• 2 Standard Deviations from the Mean: 95%
• 3 Standard Deviations from the Mean: 99.7%

Here we can calculate the outliers on the basis of 2 Std dev.

So range = Mean (of the population) ± 2*Std Dev = 105 ± 2* 13.34 = 131.67 to 78.32.

One of the reading is 74.9 which doesn't lies in the range specified above, so we can assume that as an outlier and that data point can be eliminated.

Now new data is

91.9
97.8
111.4
122.3	Mean	103.13
105.4	Std. Dev	9.68
95.0
103.8
99.6
119.3
104.8
101.7
92.1
97.6
111.1
125.9
105.4
95.0
103.7
99.7
119.1
98.1
105.0
101.6
92.2
97.6
91.60
111.5
122.20
104.90
95.50
103.50
101.10

Since, n > 30 , we can use z score.

Now to calculate , z score we have,

(x- µ )/ (σ / n^0.5) = (103.13 - 105) /(9.68/32^0.5) = -1.09

p value (using the table) = The two-tailed P value equals 0.2757

For calculating interval at 95 % CI , we have z (at 95% CI) = 1.96

Intervals at 95 confidence interval =  µ ± 1.96*σ/ n^0.5 = 105 ± 1.96*{9.68/(32^0.5)} =  108.3 to 101.07