Question

Researchers in a 2015 study by Flannery et al. sought to estimate the effectiveness of the seasonal influenza vaccine in preventing acute respiratory infection (ARI). A total of 507 subjects from Texas who showed symptoms of ARI were recruited for this study. The data provided show the result of each subject's test for influenza.

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Suppose the researchers wish to construct a 95%  z ‑confidence interval for the proportion of patients with ARI symptoms who would test positive for influenza. Calculate the lower and upper bounds of this confidence interval. Please specify your answer as a decimal, rounded to at least three decimal places.

lower bound:

upper bound:

Patient" "Influenza"
"1" "Positive"
"2" "Negative"
"3" "Positive"
"4" "Positive"
"5" "Negative"
"6" "Negative"
"7" "Positive"
"8" "Negative"
"9" "Negative"
"10" "Negative"
"11" "Positive"
"12" "Negative"
"13" "Positive"
"14" "Positive"
"15" "Positive"
"16" "Negative"
"17" "Negative"
"18" "Positive"
"19" "Positive"
"20" "Positive"
"21" "Positive"
"22" "Positive"
"23" "Positive"
"24" "Negative"
"25" "Positive"
"26" "Negative"
"27" "Positive"
"28" "Positive"
"29" "Negative"
"30" "Positive"
"31" "Positive"
"32" "Negative"
"33" "Positive"
"34" "Negative"
"35" "Negative"
"36" "Negative"
"37" "Negative"
"38" "Positive"
"39" "Positive"
"40" "Positive"
"41" "Negative"
"42" "Negative"
"43" "Negative"
"44" "Negative"
"45" "Negative"
"46" "Negative"
"47" "Negative"
"48" "Negative"
"49" "Positive"
"50" "Positive"
"51" "Positive"
"52" "Positive"
"53" "Positive"
"54" "Negative"
"55" "Negative"
"56" "Positive"
"57" "Negative"
"58" "Positive"
"59" "Positive"
"60" "Negative"
"61" "Negative"
"62" "Positive"
"63" "Negative"
"64" "Negative"
"65" "Negative"
"66" "Positive"
"67" "Positive"
"68" "Positive"
"69" "Positive"
"70" "Positive"
"71" "Positive"
"72" "Negative"
"73" "Negative"
"74" "Positive"
"75" "Negative"
"76" "Negative"
"77" "Negative"
"78" "Negative"
"79" "Negative"
"80" "Negative"
"81" "Positive"
"82" "Negative"
"83" "Negative"
"84" "Positive"
"85" "Negative"
"86" "Positive"
"87" "Positive"
"88" "Negative"
"89" "Positive"
"90" "Positive"
"91" "Negative"
"92" "Positive"
"93" "Negative"
"94" "Positive"
"95" "Negative"
"96" "Positive"
"97" "Negative"
"98" "Negative"
"99" "Negative"
"100" "Negative"
"101" "Negative"
"102" "Positive"
"103" "Positive"
"104" "Negative"
"105" "Negative"
"106" "Negative"
"107" "Positive"
"108" "Negative"
"109" "Negative"
"110" "Negative"
"111" "Negative"
"112" "Positive"
"113" "Negative"
"114" "Negative"
"115" "Negative"
"116" "Positive"
"117" "Positive"
"118" "Positive"
"119" "Negative"
"120" "Negative"
"121" "Negative"
"122" "Negative"
"123" "Negative"
"124" "Negative"
"125" "Negative"
"126" "Positive"
"127" "Negative"
"128" "Positive"
"129" "Negative"
"130" "Positive"
"131" "Negative"
"132" "Negative"
"133" "Negative"
"134" "Negative"
"135" "Negative"
"136" "Positive"
"137" "Negative"
"138" "Positive"
"139" "Negative"
"140" "Negative"
"141" "Positive"
"142" "Positive"
"143" "Negative"
"144" "Positive"
"145" "Positive"
"146" "Positive"
"147" "Negative"
"148" "Negative"
"149" "Negative"
"150" "Negative"
"151" "Positive"
"152" "Positive"
"153" "Negative"
"154" "Negative"
"155" "Negative"
"156" "Positive"
"157" "Positive"
"158" "Positive"
"159" "Positive"
"160" "Negative"
"161" "Positive"
"162" "Negative"
"163" "Positive"
"164" "Negative"
"165" "Negative"
"166" "Positive"
"167" "Negative"
"168" "Negative"
"169" "Positive"
"170" "Negative"
"171" "Positive"
"172" "Positive"
"173" "Negative"
"174" "Negative"
"175" "Negative"
"176" "Positive"
"177" "Negative"
"178" "Negative"
"179" "Positive"
"180" "Positive"
"181" "Negative"
"182" "Negative"
"183" "Negative"
"184" "Negative"
"185" "Negative"
"186" "Negative"
"187" "Negative"
"188" "Negative"
"189" "Negative"
"190" "Positive"
"191" "Positive"
"192" "Negative"
"193" "Positive"
"194" "Negative"
"195" "Negative"
"196" "Negative"
"197" "Positive"
"198" "Positive"
"199" "Positive"
"200" "Positive"
"201" "Negative"
"202" "Positive"
"203" "Negative"
"204" "Negative"
"205" "Negative"
"206" "Negative"
"207" "Positive"
"208" "Negative"
"209" "Negative"
"210" "Negative"
"211" "Positive"
"212" "Negative"
"213" "Positive"
"214" "Negative"
"215" "Negative"
"216" "Negative"
"217" "Negative"
"218" "Positive"
"219" "Positive"
"220" "Negative"
"221" "Positive"
"222" "Positive"
"223" "Negative"
"224" "Positive"
"225" "Negative"
"226" "Negative"
"227" "Negative"
"228" "Negative"
"229" "Positive"
"230" "Negative"
"231" "Negative"
"232" "Negative"
"233" "Positive"
"234" "Positive"
"235" "Positive"
"236" "Negative"
"237" "Positive"
"238" "Negative"
"239" "Negative"
"240" "Negative"
"241" "Positive"
"242" "Negative"
"243" "Negative"
"244" "Positive"
"245" "Negative"
"246" "Negative"
"247" "Negative"
"248" "Negative"
"249" "Negative"
"250" "Positive"
"251" "Positive"
"252" "Positive"
"253" "Negative"
"254" "Negative"
"255" "Positive"
"256" "Negative"
"257" "Negative"
"258" "Negative"
"259" "Negative"
"260" "Positive"
"261" "Negative"
"262" "Negative"
"263" "Negative"
"264" "Positive"
"265" "Positive"
"266" "Negative"
"267" "Negative"
"268" "Negative"
"269" "Negative"
"270" "Negative"
"271" "Positive"
"272" "Positive"
"273" "Negative"
"274" "Negative"
"275" "Negative"
"276" "Negative"
"277" "Positive"
"278" "Negative"
"279" "Negative"
"280" "Negative"
"281" "Positive"
"282" "Negative"
"283" "Negative"
"284" "Negative"
"285" "Positive"
"286" "Positive"
"287" "Positive"
"288" "Negative"
"289" "Negative"
"290" "Negative"
"291" "Negative"
"292" "Positive"
"293" "Negative"
"294" "Negative"
"295" "Positive"
"296" "Negative"
"297" "Negative"
"298" "Positive"
"299" "Negative"
"300" "Positive"
"301" "Negative"
"302" "Positive"
"303" "Positive"
"304" "Positive"
"305" "Negative"
"306" "Negative"
"307" "Negative"
"308" "Positive"
"309" "Negative"
"310" "Negative"
"311" "Negative"
"312" "Negative"
"313" "Negative"
"314" "Positive"
"315" "Negative"
"316" "Negative"
"317" "Negative"
"318" "Negative"
"319" "Positive"
"320" "Negative"
"321" "Positive"
"322" "Negative"
"323" "Positive"
"324" "Negative"
"325" "Positive"
"326" "Negative"
"327" "Positive"
"328" "Negative"
"329" "Negative"
"330" "Negative"
"331" "Positive"
"332" "Positive"
"333" "Negative"
"334" "Negative"
"335" "Negative"
"336" "Positive"
"337" "Negative"
"338" "Negative"
"339" "Negative"
"340" "Positive"
"341" "Negative"
"342" "Negative"
"343" "Positive"
"344" "Negative"
"345" "Positive"
"346" "Negative"
"347" "Negative"
"348" "Positive"
"349" "Positive"
"350" "Negative"
"351" "Positive"
"352" "Negative"
"353" "Negative"
"354" "Positive"
"355" "Positive"
"356" "Positive"
"357" "Negative"
"358" "Negative"
"359" "Negative"
"360" "Negative"
"361" "Positive"
"362" "Negative"
"363" "Positive"
"364" "Positive"
"365" "Positive"
"366" "Positive"
"367" "Negative"
"368" "Positive"
"369" "Negative"
"370" "Negative"
"371" "Negative"
"372" "Negative"
"373" "Positive"
"374" "Negative"
"375" "Negative"
"376" "Negative"
"377" "Negative"
"378" "Negative"
"379" "Positive"
"380" "Positive"
"381" "Negative"
"382" "Positive"
"383" "Negative"
"384" "Positive"
"385" "Positive"
"386" "Negative"
"387" "Negative"
"388" "Positive"
"389" "Negative"
"390" "Negative"
"391" "Positive"
"392" "Negative"
"393" "Positive"
"394" "Positive"
"395" "Negative"
"396" "Positive"
"397" "Positive"
"398" "Negative"
"399" "Negative"
"400" "Positive"
"401" "Positive"
"402" "Negative"
"403" "Positive"
"404" "Negative"
"405" "Positive"
"406" "Negative"
"407" "Positive"
"408" "Positive"
"409" "Negative"
"410" "Negative"
"411" "Negative"
"412" "Positive"
"413" "Positive"
"414" "Positive"
"415" "Positive"
"416" "Positive"
"417" "Negative"
"418" "Positive"
"419" "Negative"
"420" "Negative"
"421" "Positive"
"422" "Positive"
"423" "Negative"
"424" "Negative"
"425" "Negative"
"426" "Negative"
"427" "Positive"
"428" "Negative"
"429" "Negative"
"430" "Negative"
"431" "Negative"
"432" "Negative"
"433" "Negative"
"434" "Positive"
"435" "Negative"
"436" "Negative"
"437" "Negative"
"438" "Negative"
"439" "Negative"
"440" "Negative"
"441" "Positive"
"442" "Positive"
"443" "Positive"
"444" "Positive"
"445" "Positive"
"446" "Negative"
"447" "Positive"
"448" "Positive"
"449" "Negative"
"450" "Positive"
"451" "Negative"
"452" "Negative"
"453" "Negative"
"454" "Positive"
"455" "Negative"
"456" "Negative"
"457" "Positive"
"458" "Positive"
"459" "Positive"
"460" "Positive"
"461" "Negative"
"462" "Negative"
"463" "Positive"
"464" "Positive"
"465" "Negative"
"466" "Negative"
"467" "Negative"
"468" "Negative"
"469" "Negative"
"470" "Negative"
"471" "Negative"
"472" "Positive"
"473" "Negative"
"474" "Positive"
"475" "Positive"
"476" "Positive"
"477" "Positive"
"478" "Positive"
"479" "Positive"
"480" "Negative"
"481" "Negative"
"482" "Negative"
"483" "Positive"
"484" "Negative"
"485" "Negative"
"486" "Negative"
"487" "Negative"
"488" "Negative"
"489" "Positive"
"490" "Positive"
"491" "Negative"
"492" "Positive"
"493" "Negative"
"494" "Negative"
"495" "Positive"
"496" "Positive"
"497" "Negative"
"498" "Negative"
"499" "Negative"
"500" "Positive"
"501" "Positive"
"502" "Negative"
"503" "Negative"
"504" "Positive"
"505" "Positive"
"506" "Negative"
"507" "Positive"

Answer 1

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Output from minitab is below :

Descriptive Statistics

N	Event	Sample p	95% CI for p
507	210	0.414201	(0.371324, 0.457078)

Below is done manually :